Abdul-Majid Wazwaz
Linear and Nonlinear Integral Equations Methods and Applications
Abdul-Majid Wazwaz
Linear and N...

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Abdul-Majid Wazwaz

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Abdul-Majid Wazwaz

Linear and Nonlinear Integral Equations Methods and Applications

Abdul-Majid Wazwaz

Linear and Nonlinear Integral Equations Methods and Applications

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THIS BOOK IS DEDICATED TO my wife, our son, and our three daughters for supporting me in all my endeavors

Preface

Many remarkable advances have been made in the ﬁeld of integral equations, but these remarkable developments have remained scattered in a variety of specialized journals. These new ideas and approaches have rarely been brought together in textbook form. If these ideas merely remain in scholarly journals and never get discussed in textbooks, then specialists and students will not be able to beneﬁt from the results of the valuable research achievements. The explosive growth in industry and technology requires constructive adjustments in mathematics textbooks. The valuable achievements in research work are not found in many of today’s textbooks, but they are worthy of addition and study. The technology is moving rapidly, which is pushing for valuable insights into some substantial applications and developed approaches. The mathematics taught in the classroom should come to resemble the mathematics used in varied applications of nonlinear science models and engineering applications. This book was written with these thoughts in mind. Linear and Nonlinear Integral Equations: Methods and Applications is designed to serve as a text and a reference. The book is designed to be accessible to advanced undergraduate and graduate students as well as a research monograph to researchers in applied mathematics, physical sciences, and engineering. This text diﬀers from other similar texts in a number of ways. First, it explains the classical methods in a comprehensible, non-abstract approach. Furthermore, it introduces and explains the modern developed mathematical methods in such a fashion that shows how the new methods can complement the traditional methods. These approaches further improve the understanding of the material. The book avoids approaching the subject through the compact and classical methods that make the material diﬃcult to be grasped, especially by students who do not have the background in these abstract concepts. The aim of this book is to oﬀer practical treatment of linear and nonlinear integral equations emphasizing the need to problem solving rather than theorem proving. The book was developed as a result of many years of experiences in teaching integral equations and conducting research work in this ﬁeld. The author

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Preface

has taken account of his teaching experience, research work as well as valuable suggestions received from students and scholars from a wide variety of audience. Numerous examples and exercises, ranging in level from easy to difﬁcult, but consistent with the material, are given in each section to give the reader the knowledge, practice and skill in linear and nonlinear integral equations. There is plenty of material in this text to be covered in two semesters for senior undergraduates and beginning graduates of mathematics, physical science, and engineering. The content of the book is divided into two distinct parts, and each part is self-contained and practical. Part I contains twelve chapters that handle the linear integral and nonlinear integro-diﬀerential equations by using the modern mathematical methods, and some of the powerful traditional methods. Since the book’s readership is a diverse and interdisciplinary audience of applied mathematics, physical science, and engineering, attempts are made so that part I presents both analytical and numerical approaches in a clear and systematic fashion to make this book accessible to those who work in these ﬁelds. Part II contains the remaining six chapters devoted to thoroughly examining the nonlinear integral equations and its applications. The potential theory contributed more than any ﬁeld to give rise to nonlinear integral equations. Mathematical physics models, such as diﬀraction problems, scattering in quantum mechanics, conformal mapping, and water waves also contributed to the creation of nonlinear integral equations. Because it is not always possible to ﬁnd exact solutions to problems of physical science that are posed, much work is devoted to obtaining qualitative approximations that highlight the structure of the solution. Chapter 1 provides the basic deﬁnitions and introductory concepts. The Taylor series, Leibnitz rule, and Laplace transform method are presented and reviewed. This discussion will provide the reader with a strong basis to understand the thoroughly-examined material in the following chapters. In Chapter 2, the classiﬁcations of integral and integro-diﬀerential equations are presented and illustrated. In addition, the linearity and the homogeneity concepts of integral equations are clearly addressed. The conversion process of IVP and BVP to Volterra integral equation and Fredholm integral equation respectively are described. Chapters 3 and 5 deal with the linear Volterra integral equations and the linear Volterra integro-diﬀerential equations, of the ﬁrst and the second kind, respectively. Each kind is approached by a variety of methods that are described in details. Chapters 3 and 5 provide the reader with a comprehensive discussion of both types of equations. The two chapters emphasize the power of the proposed methods in handling these equations. Chapters 4 and 6 are entirely devoted to Fredholm integral equations and Fredholm integro-diﬀerential equations, of the ﬁrst and the second kind, respectively. The ill-posed Fredholm integral equation of the ﬁrst kind is handled by the powerful method of regularization combined with other methods. The two kinds of equations are approached

Preface

ix

by many appropriate algorithms that are illustrated in details. A comprehensive study is introduced where a variety of reliable methods is applied independently and supported by many illustrative examples. Chapter 7 is devoted to studying the Abel’s integral equations, generalized Abel’s integral equations, and the weakly singular integral equations. The chapter also stresses the signiﬁcant features of these types of singular equations with full explanations and many illustrative examples and exercises. Chapters 8 and 9 introduce a valuable study on Volterra-Fredholm integral equations and Volterra-Fredholm integro-diﬀerential equations respectively in one and two variables. The mixed Volterra-Fredholm integral and the mixed VolterraFredholm integro-diﬀerential equations in two variables are also examined with illustrative examples. The proposed methods introduce a powerful tool for handling these two types of equations. Examples are provided with a substantial amount of explanation. The reader will ﬁnd a wealth of well-known models with one and two variables. A detailed and clear explanation of every application is introduced and supported by fully explained examples and exercises of every type. Chapters 10, 11, and 12 are concerned with the systems of Volterra integral and integro-diﬀerential equations, systems of Fredholm integral and integro-diﬀerential equations, and systems of singular integral equations and systems of weakly singular integral equations respectively. Systems of integral equations that are important, are handled by using very constructive methods. A discussion of the basic theory and illustrations of the solutions to the systems are presented to introduce the material in a clear and useful fashion. Singular systems in one, two, and three variables are thoroughly investigated. The systems are supported by a variety of useful methods that are well explained and illustrated. Part II is titled “Nonlinear Integral Equations”. Part II of this book gives a self-contained, practical and realistic approach to nonlinear integral equations, where scientists and engineers are paying great attention to the eﬀects caused by the nonlinearity of dynamical equations in nonlinear science. The potential theory contributed more than any ﬁeld to give rise to nonlinear integral equations. Mathematical physics models, such as diﬀraction problems, scattering in quantum mechanics, conformal mapping, and water waves also contributed to the creation of nonlinear integral equations. The nonlinearity of these models may give more than one solution and this is the nature of nonlinear problems. Moreover, ill-posed Fredholm integral equations of the ﬁrst kind may also give more than one solution even if it is linear. Chapter 13 presents discussions about nonlinear Volterra integral equations and systems of Volterra integral equations, of the ﬁrst and the second kind. More emphasis on the existence of solutions is proved and emphasized. A variety of methods are employed, introduced and explained in a clear and useful manner. Chapter 14 is devoted to giving a comprehensive study on nonlinear Volterra integro-diﬀerential equations and systems of nonlinear Volterra integro-diﬀerential equations, of the ﬁrst and the second kind.

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Preface

A variety of methods are introduced, and numerous practical examples are explained in a practical way. Chapter 15 investigates thoroughly the existence theorem, bifurcation points and singular points that may arise from nonlinear Fredholm integral equations. The study presents a variety of powerful methods to handle nonlinear Fredholm integral equations of the ﬁrst and the second kind. Systems of these equations are examined with illustrated examples. Chapter 16 is entirely devoted to studying a family of nonlinear Fredholm integro-diﬀerential equations of the second kind and the systems of these equations. The approach we followed is identical to our approach in the previous chapters to make the discussion accessible for interdisciplinary audience. Chapter 17 provides the reader with a comprehensive discussion of the nonlinear singular integral equations, nonlinear weakly singular integral equations, and systems of these equations. Most of these equations are characterized by the singularity behavior where the proposed methods should overcome the diﬃculty of this singular behavior. The power of the employed methods is conﬁrmed here by determining solutions that may not be unique. Chapter 18 presents a comprehensive study on ﬁve scientiﬁc applications that we selected because of its wide applicability for several other models. Because it is not always possible to ﬁnd exact solutions to models of physical sciences, much work is devoted to obtaining qualitative approximations that highlight the structure of the solution. The powerful Pad´e approximants are used to give insight into the structure of the solution. This chapter closes Part II of this text. The book concludes with seven useful appendices. Moreover, the book introduces the traditional methods in the same amount of concern to provide the reader with the knowledge needed to make a comparison. I deeply acknowledge Professor Albert Luo for many helpful discussions, encouragement, and useful remarks. I am also indebted to Ms. Liping Wang, the Publishing Editor of the Higher Education Press for her eﬀective cooperation and important suggestions. The staﬀ of HEP deserve my thanks for their support to this project. I owe them all my deepest thanks. I also deeply acknowledge Professor Louis Pennisi who made very valuable suggestions that helped a great deal in directing this book towards its main goal. I am deeply indebted to my wife, my son and my daughters who provided me with their continued encouragement, patience and support during the long days of preparing this book. The author would highly appreciate any notes concerning any constructive suggestions.

Abdul-Majid Wazwaz Saint Xavier University Chicago, IL 60655 April 20, 2011

Contents

Part I Linear Integral Equations 1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Ordinary Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 First Order Linear Diﬀerential Equations . . . . . . . . . . . . 1.2.2 Second Order Linear Diﬀerential Equations . . . . . . . . . . 1.2.3 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 1.3 Leibnitz Rule for Diﬀerentiation of Integrals . . . . . . . . . . . . . . . 1.4 Reducing Multiple Integrals to Single Integrals . . . . . . . . . . . . . 1.5 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Properties of Laplace Transforms . . . . . . . . . . . . . . . . . . . 1.6 Inﬁnite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 4 7 7 9 13 17 19 22 23 28 30

2

Introductory Concepts of Integral Equations . . . . . . . . . . . . . . 2.1 Classiﬁcation of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Fredholm Integral Equations . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Volterra-Fredholm Integral Equations . . . . . . . . . . . . . . . 2.1.4 Singular Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Classiﬁcation of Integro-Diﬀerential Equations . . . . . . . . . . . . . 2.2.1 Fredholm Integro-Diﬀerential Equations . . . . . . . . . . . . . 2.2.2 Volterra Integro-Diﬀerential Equations . . . . . . . . . . . . . . 2.2.3 Volterra-Fredholm Integro-Diﬀerential Equations . . . . . 2.3 Linearity and Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Linearity Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Homogeneity Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Origins of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Converting IVP to Volterra Integral Equation . . . . . . . . . . . . . .

33 34 34 35 35 36 37 38 38 39 40 40 41 42 42

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Contents

2.5.1 Converting Volterra Integral Equation to IVP . . . . . . . . 2.6 Converting BVP to Fredholm Integral Equation . . . . . . . . . . . . 2.6.1 Converting Fredholm Integral Equation to BVP . . . . . . 2.7 Solution of an Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 49 54 59 63

Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Volterra Integral Equations of the Second Kind . . . . . . . . . . . . 3.2.1 The Adomian Decomposition Method . . . . . . . . . . . . . . . 3.2.2 The Modiﬁed Decomposition Method . . . . . . . . . . . . . . . 3.2.3 The Noise Terms Phenomenon . . . . . . . . . . . . . . . . . . . . . 3.2.4 The Variational Iteration Method . . . . . . . . . . . . . . . . . . 3.2.5 The Successive Approximations Method . . . . . . . . . . . . . 3.2.6 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . 3.2.7 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 3.3 Volterra Integral Equations of the First Kind . . . . . . . . . . . . . . 3.3.1 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . 3.3.3 Conversion to a Volterra Equation of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66 66 73 78 82 95 99 103 108 108 111

4

Fredholm Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fredholm Integral Equations of the Second Kind . . . . . . . . . . . 4.2.1 The Adomian Decomposition Method . . . . . . . . . . . . . . . 4.2.2 The Modiﬁed Decomposition Method . . . . . . . . . . . . . . . 4.2.3 The Noise Terms Phenomenon . . . . . . . . . . . . . . . . . . . . . 4.2.4 The Variational Iteration Method . . . . . . . . . . . . . . . . . . 4.2.5 The Direct Computation Method . . . . . . . . . . . . . . . . . . . 4.2.6 The Successive Approximations Method . . . . . . . . . . . . . 4.2.7 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 4.3 Homogeneous Fredholm Integral Equation . . . . . . . . . . . . . . . . . 4.3.1 The Direct Computation Method . . . . . . . . . . . . . . . . . . . 4.4 Fredholm Integral Equations of the First Kind . . . . . . . . . . . . . 4.4.1 The Method of Regularization . . . . . . . . . . . . . . . . . . . . . 4.4.2 The Homotopy Perturbation Method . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 121 121 128 133 136 141 146 151 154 155 159 161 166 173

5

Volterra Integro-Diﬀerential Equations . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Volterra Integro-Diﬀerential Equations of the Second Kind . . . 5.2.1 The Adomian Decomposition Method . . . . . . . . . . . . . . . 5.2.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . .

175 175 176 176 181

3

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Contents

5.2.3 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . 5.2.4 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Converting Volterra Integro-Diﬀerential Equations to Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Converting Volterra Integro-Diﬀerential Equation to Volterra Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Volterra Integro-Diﬀerential Equations of the First Kind . . . . . 5.3.1 Laplace Transform Method . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

7

8

Fredholm Integro-Diﬀerential Equations . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 The Direct Computation Method . . . . . . . . . . . . . . . . . . . 6.2.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . . 6.2.3 The Adomian Decomposition Method . . . . . . . . . . . . . . . 6.2.4 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abel’s Integral Equation and Singular Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Abel’s Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . 7.3 The Generalized Abel’s Integral Equation . . . . . . . . . . . . . . . . . 7.3.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . 7.3.2 The Main Generalized Abel Equation . . . . . . . . . . . . . . . 7.4 The Weakly Singular Volterra Equations . . . . . . . . . . . . . . . . . . 7.4.1 The Adomian Decomposition Method . . . . . . . . . . . . . . . 7.4.2 The Successive Approximations Method . . . . . . . . . . . . . 7.4.3 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volterra-Fredholm Integral Equations . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Volterra-Fredholm Integral Equations . . . . . . . . . . . . . . . . . 8.2.1 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 The Adomian Decomposition Method . . . . . . . . . . . . . . . 8.3 The Mixed Volterra-Fredholm Integral Equations . . . . . . . . . . . 8.3.1 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 The Adomian Decomposition Method . . . . . . . . . . . . . . . 8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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186 190 195 199 203 204 206 211 213 213 214 214 218 223 230 234 237 237 238 239 242 243 245 247 248 253 257 260 261 261 262 262 266 269 270 273 277

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8.4.1 The Modiﬁed Decomposition Method . . . . . . . . . . . . . . . 278 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Volterra-Fredholm Integro-Diﬀerential Equations . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Volterra-Fredholm Integro-Diﬀerential Equation . . . . . . . . 9.2.1 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . . 9.3 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 The Direct Computation Method . . . . . . . . . . . . . . . . . . . 9.3.2 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 9.4 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 The Modiﬁed Decomposition Method . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

285 285 285 285 289

10 Systems of Volterra Integral Equations . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Systems of Volterra Integral Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 The Adomian Decomposition Method . . . . . . . . . . . . . 10.2.2 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . 10.3 Systems of Volterra Integral Equations of the First Kind . . . 10.3.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . 10.3.2 Conversion to a Volterra System of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Systems of Volterra Integro-Diﬀerential Equations . . . . . . . . . 10.4.1 The Variational Iteration Method . . . . . . . . . . . . . . . . . 10.4.2 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311 311

11 Systems of Fredholm Integral Equations . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Systems of Fredholm Integral Equations . . . . . . . . . . . . . . . . . . 11.2.1 The Adomian Decomposition Method . . . . . . . . . . . . . 11.2.2 The Direct Computation Method . . . . . . . . . . . . . . . . . 11.3 Systems of Fredholm Integro-Diﬀerential Equations . . . . . . . . 11.3.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 11.3.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341 341 342 342 347 352 353 358 364

9

296 296 300 303 304 309

312 312 318 323 323 327 328 329 335 339

Contents

12 Systems of Singular Integral Equations . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Systems of Generalized Abel Integral Equations . . . . . . . . . . . 12.2.1 Systems of Generalized Abel Integral Equations in Two Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Systems of Generalized Abel Integral Equations in Three Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Systems of the Weakly Singular Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . 12.3.2 The Adomian Decomposition Method . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

365 365 366 366 370 374 374 378 383

Part II Nonlinear Integral Equations 13 Nonlinear Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Existence of the Solution for Nonlinear Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Nonlinear Volterra Integral Equations of the Second Kind . . 13.3.1 The Successive Approximations Method . . . . . . . . . . . 13.3.2 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . 13.3.3 The Adomian Decomposition Method . . . . . . . . . . . . . 13.4 Nonlinear Volterra Integral Equations of the First Kind . . . . 13.4.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . 13.4.2 Conversion to a Volterra Equation of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Systems of Nonlinear Volterra Integral Equations . . . . . . . . . . 13.5.1 Systems of Nonlinear Volterra Integral Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Systems of Nonlinear Volterra Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Nonlinear Volterra Integro-Diﬀerential Equations . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 The Combined Laplace Transform-Adomian Decomposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . 14.2.3 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . 14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

387 387 388 388 389 393 397 404 405 408 411 412 417 423 425 425 426 426 432 436 440

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Contents

14.3.1 The Combined Laplace Transform-Adomian Decomposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Conversion to Nonlinear Volterra Equation of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 The Variational Iteration Method . . . . . . . . . . . . . . . . . 14.4.2 The Combined Laplace Transform-Adomian Decomposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

440 446 450 451 456 465

15 Nonlinear Fredholm Integral Equations . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Existence of the Solution for Nonlinear Fredholm Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Bifurcation Points and Singular Points . . . . . . . . . . . . . 15.3 Nonlinear Fredholm Integral Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 15.3.2 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . 15.3.3 The Adomian Decomposition Method . . . . . . . . . . . . . 15.3.4 The Successive Approximations Method . . . . . . . . . . . 15.4 Homogeneous Nonlinear Fredholm Integral Equations . . . . . . 15.4.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 15.5 Nonlinear Fredholm Integral Equations of the First Kind . . . 15.5.1 The Method of Regularization . . . . . . . . . . . . . . . . . . . . 15.5.2 The Homotopy Perturbation Method . . . . . . . . . . . . . . 15.6 Systems of Nonlinear Fredholm Integral Equations . . . . . . . . . 15.6.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 15.6.2 The Modiﬁed Adomian Decomposition Method . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

467 467

16 Nonlinear Fredholm Integro-Diﬀerential Equations . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Nonlinear Fredholm Integro-Diﬀerential Equations . . . . . . . . . 16.2.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 16.2.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . 16.2.3 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . 16.3 Homogeneous Nonlinear Fredholm Integro-Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 16.4 Systems of Nonlinear Fredholm Integro-Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 16.4.2 The Variational Iteration Method . . . . . . . . . . . . . . . . .

517 517 518 518 522 526

468 469 469 470 476 480 485 490 490 494 495 500 505 506 510 515

530 530 535 535 540

Contents

xvii

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 17 Nonlinear Singular Integral Equations . . . . . . . . . . . . . . . . . . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Nonlinear Abel’s Integral Equation . . . . . . . . . . . . . . . . . . . . . . 17.2.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . 17.3 The Generalized Nonlinear Abel Equation . . . . . . . . . . . . . . . . 17.3.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . 17.3.2 The Main Generalized Nonlinear Abel Equation . . . . 17.4 The Nonlinear Weakly-Singular Volterra Equations . . . . . . . . 17.4.1 The Adomian Decomposition Method . . . . . . . . . . . . . 17.5 Systems of Nonlinear Weakly-Singular Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5.1 The Modiﬁed Adomian Decomposition Method . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

547 547 548 549 552 553 556 559 559

18 Applications of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Volterra’s Population Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 The Variational Iteration Method . . . . . . . . . . . . . . . . . 18.2.2 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . 18.2.3 The Pad´e Approximants . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Integral Equations with Logarithmic Kernels . . . . . . . . . . . . . . 18.3.1 Second Kind Fredholm Integral Equation with a Logarithmic Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.2 First Kind Fredholm Integral Equation with a Logarithmic Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.3 Another First Kind Fredholm Integral Equation with a Logarithmic Kernel . . . . . . . . . . . . . . . . . . . . . . . 18.4 The Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5 The Thomas-Fermi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6 Heat Transfer and Heat Radiation . . . . . . . . . . . . . . . . . . . . . . . 18.6.1 Heat Transfer: Lighthill Singular Integral Equation . . 18.6.2 Heat Radiation in a Semi-Inﬁnite Solid . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

569 569 570 571 572 573 574

583 584 587 590 590 592 594

Appendix A Table of Indeﬁnite Integrals . . . . . . . . . . . . . . . . . . . . A.1 Basic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Trigonometric Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Inverse Trigonometric Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Exponential and Logarithmic Forms . . . . . . . . . . . . . . . . . . . . . . A.5 Hyperbolic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Other Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

597 597 597 598 598 599 599

562 563 567

577 580

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Contents

Appendix B B.1

Integrals Involving Irrational Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 n Integrals Involving √tx−t , n is an integer, n 0 . . . . . . . . . . . . 600

B.2

Integrals Involving

n

2 √t , x−t

n is an odd integer, n 1 . . . . . . . . 600

Appendix C Series Representations . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Exponential Functions Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . C.4 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5 Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.6 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

601 601 601 602 602 602 602

Appendix D The Error and the Complementary Error Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 D.1 The Error Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 D.2 The Complementary Error Function . . . . . . . . . . . . . . . . . . . . . . 603 Appendix E

Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

Appendix F Inﬁnite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 F.1 Numerical Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 F.2 Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Appendix G The Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 G.1 The Fresnel Cosine Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 G.2 The Fresnel Sine Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

Part I

Linear Integral Equations

Chapter 1

Preliminaries

An integral equation is an equation in which the unknown function u(x) appears under an integral sign [1–7]. A standard integral equation in u(x) is of the form: h(x)

u(x) = f (x) + λ

K(x, t)u(t)dt,

(1.1)

g(x)

where g(x) and h(x) are the limits of integration, λ is a constant parameter, and K(x, t) is a function of two variables x and t called the kernel or the nucleus of the integral equation. The function u(x) that will be determined appears under the integral sign, and it appears inside the integral sign and outside the integral sign as well. The functions f (x) and K(x, t) are given in advance. It is to be noted that the limits of integration g(x) and h(x) may be both variables, constants, or mixed. An integro-diﬀerential equation is an equation in which the unknown function u(x) appears under an integral sign and contains an ordinary derivative u(n) (x) as well. A standard integro-diﬀerential equation is of the form: h(x) K(x, t)u(t)dt, (1.2) u(n) (x) = f (x) + λ g(x)

where g(x), h(x), f (x), λ and the kernel K(x, t) are as prescribed before. Integral equations and integro-diﬀerential equations will be classiﬁed into distinct types according to the limits of integration and the kernel K(x, t). All types of integral equations and integro-diﬀerential equations will be classiﬁed and investigated in the forthcoming chapters. In this chapter, we will review the most important concepts needed to study integral equations. The traditional methods, such as Taylor series method and the Laplace transform method, will be used in this text. Moreover, the recently developed methods, that will be used thoroughly in this text, will determine the solution in a power series that will converge to an exact solution if such a solution exists. However, if exact solution does not exist, we use as many terms of the obtained series for numerical purposes to approximate the solution. The more terms we determine the higher numerical A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

4

1 Preliminaries

accuracy we can achieve. Furthermore, we will review the basic concepts for solving ordinary diﬀerential equations. Other mathematical concepts, such as Leibnitz rule will be presented.

1.1 Taylor Series Let f (x) be a function with derivatives of all orders in an interval [x0 , x1 ] that contains an interior point a. The Taylor series of f (x) generated at x = a is ∞ f (n) (a) (x − a)n , (1.3) f (x) = n! n=0 or equivalently f (a) f (a) f (a) (x − a) + (x − a)2 + (x − a)3 + · · · 1! 2! 3! (1.4) f (n) (a) (x − a)n + · · · . + n! The Taylor series generated by f (x) at a = 0 is called the Maclaurin series and given by ∞ f (n) (0) n x , f (x) = (1.5) n! n=0 f (x) = f (a) +

that is equivalent to f (0) 2 f (0) 3 f (n) (0) n f (0) x+ x + x +···+ x + · · · . (1.6) f (x) = f (0) + 1! 2! 3! n! In what follows, we will discuss a few examples for the determination of the Taylor series at x = 0. Example 1.1 Find the Taylor series generated by f (x) = ex at x = 0. We list the exponential function and its derivatives as follows: f (n) (x) f (x) = ex f (x) = ex f (x) = ex f (x) = ex f (n) (0) f (0) = 1, f (0) = 1, f (0) = 1, .. . and so on. This gives the Taylor series for ex by x3 x4 x2 + + + ··· ex = 1 + x + 2! 3! 4! and in a compact form by ∞ xn ex = . n! n=0 Similarly, we can easily show that

f (0) = 1,

(1.7)

(1.8)

1.1 Taylor Series

5

x3 x4 x2 − + + ··· 2! 3! 4!

(1.9)

(ax)3 (ax)4 (ax)2 + + + ··· 2! 3! 4!

(1.10)

e−x = 1 − x + and eax = 1 + ax + Example 1.2

Find the Taylor series generated by f (x) = cos x at x = 0. Following the discussions presented before we ﬁnd f (n) (x) f (x) = cos x,f (x) = − sin x,f (x) = − cos x,f (x) = sin x,f (iv) (x) = cos x f (n) (0) f (0) = 1, .. .

f (0) = 0,

f (0) = −1,

f (0) = 0,

f (iv) (0) = 1,

and so on. This gives the Taylor series for cos x by x4 x2 + + ··· cos x = 1 − 2! 4! and in a compact form by ∞ (−1)n 2n cos x = x . (2n)! n=0

(1.11)

(1.12)

In a similar way we can derive the following cos(ax) = 1 −

∞ (−1)n (ax)4 (ax)2 + + ··· = (ax)2n . 2! 4! (2n)! n=0

For f (x) = sin x and f (x) = sin(ax), we can show that ∞ (−1)n 2n+1 x3 x5 sin x = x − + + ··· = x , 3! 5! (2n + 1)! n=0 sin(ax) = (ax) −

∞ (ax)5 (−1)n (ax)3 + + ··· = (ax)2n+1 . 3! 5! (2n + 1)! n=0

(1.13)

(1.14)

In Appendix C, the Taylor series for many well known functions generated at x = 0 are given. As stated before, the newly developed methods for solving integral equations determine the solution in a series form. Unlike calculus where we determine the Taylor series for functions and the radius of convergence for each series, it is required here that we determine the exact solution of the integral equation if we determine its series solution. In what follows, we will discuss some examples that will show how exact solution is obtained if the series solution is given. Recall that the Taylor series exists for analytic functions only. Example 1.3 Find the closed form function for the following series:

6

1 Preliminaries

4 2 f (x) = 1 + 2x + 2x2 + x3 + x4 + · · · 3 3 It is obvious that this series can be rewritten in the form: (2x)3 (2x)4 (2x)2 + + + ··· f (x) = 1 + 2x + 2! 3! 4! that will converge to the exact form: f (x) = e2x .

(1.15)

(1.16) (1.17)

Example 1.4 Find the closed form function for the following series: 9 27 81 f (x) = 1 + x2 + x4 + x6 + · · · 2 8 80

(1.18)

2

, therefore the series can Notice that the second term can be written as (3x) 2! be rewritten as (3x)2 (3x)4 (3x)6 f (x) = 1 + + + + ··· (1.19) 2! 4! 6! that will converge to f (x) = cosh(3x). (1.20) Example 1.5 Find the closed form function for the following series: 1 2 f (x) = x + x3 + x5 + · · · 3 15 This series will converge to f (x) = tan x.

(1.21) (1.22)

Example 1.6 Find the closed form function for the following series: 1 1 1 1 (1.23) f (x) = 1 + x − x2 − x3 + x4 + x5 + · · · 2! 3! 4! 5! The signs of the terms are positive for the ﬁrst two terms then negative for the next two terms and so on. The series should be grouped as 1 1 1 1 (1.24) f (x) = (1 − x2 + x4 + · · · ) + (x − x3 + x5 + · · · ), 2! 4! 3! 5! that will converge to f (x) = cos x + sin x. (1.25)

Exercises 1.1 Find the closed form function for the following Taylor series: 1. f (x) = 2x + 2x2 +

4 3 2 4 9 9 27 4 x + x + · · · 2. f (x) = 1 − 3x + x2 − x3 + x + ··· 3 3 2 2 8

1.2 Ordinary Diﬀerential Equations

7

1 2 1 1 x + x3 + x4 + · · · 2! 3! 4! 1 1 1 1 1 4. f (x) = 1 − x + x2 + x3 + x4 − x5 − x6 + · · · 2! 3! 4! 5! 6! 9 81 5 243 7 5. f (x) = 3x − x3 + x − x + ··· 2 40 560 4 4 5 8 7 6. f (x) = 2x + x3 + x + x + ··· 3 15 315 2 4 6 9 27 4 81 6 8. f (x) = x2 + 7. f (x) = 1 + 2x2 + x4 + x + ··· x + x + ··· 3 45 2 8 80 2 4 6 9. f (x) = 2 − 2 x2 + x4 − x + ··· 3 45 1 5 1 1 x − x7 + · · · 10. f (x) = 1 + x − x3 + 6 120 5040 1 2 5 17 7 x + x + ··· 11. f (x) = x + x3 + 3 15 315 2 5 17 7 1 x − x + ··· 12. f (x) = x − x3 + 3 15 315 1 1 1 1 13. f (x) = 2 + 2x + x2 − x3 + x4 + x5 + · · · . 2! 3! 4! 5! 1 1 1 14. 2 + x − x2 + x4 − x6 + · · · 2! 4! 6! 3. f (x) = x +

1.2 Ordinary Diﬀerential Equations In this section we will review some of the linear ordinary diﬀerential equations that we will use for solving integral equations. For proofs, existence and uniqueness of solutions, and other details, the reader is advised to use ordinary diﬀerential equations texts.

1.2.1 First Order Linear Diﬀerential Equations The standard form of ﬁrst order linear ordinary diﬀerential equation is u + p(x)u = q(x), (1.26) where p(x) and q(x) are given continuous functions on x0 < x < x1 . We ﬁrst determine an integrating factor μ(x) by using the formula: μ(x) = e

x

p(t)dt

.

(1.27)

Recall that an integrating factor μ(x) is a function of x that is used to facilitate the solving of a diﬀerential equation. The solution of (1.26) is obtained by using the formula: x 1 u(x) = μ(t)q(t)dt + c , (1.28) μ(x)

8

1 Preliminaries

where c is an arbitrary constant that can be determined by using a given initial condition. Example 1.7 Solve the following ﬁrst order ODE: u − 3u = 3x2 e3x ,

u(0) = 1.

(1.29)

2 3x

Notice that p(x) = −3 and q(x) = 3x e . The integrating factor μ(x) is obtained by x μ(x) = e 0 −3dt = e−3x . (1.30) Consequently, u(x) is obtained by using x 1 3x u(x) = μ(t)q(t)dt + c = e μ(x)

= e3x x3 + c = e3x (x3 + 1), obtained upon using the given initial condition.

x

2

3t dt + c

(1.31)

Example 1.8 Solve the following ﬁrst order ODE: cos x , u(π) = 0, x > 0. (1.32) xu + 3u = x We ﬁrst divide the equation by x to convert it to the standard form (1.26). x . The integrating factor μ(x) is As a result, p(x) = x3 and q(x) = cos x2 μ(x) = e

x

3 t dt

= e3 ln x = x3 .

Consequently, u(x) is x

1 1 x u(x) = μ(t)q(t)dt + c = 3 t cos tdt + c μ(x) x 1 1 = 3 (cos x + x sin x + c) = 3 (cos x + x sin x + 1), x x obtained upon using the given initial condition.

(1.33)

(1.34)

Exercises 1.2.1 Find the general solution for each of the following ﬁrst order ODEs: 1. u + u = e−x , x > 0

2. xu − 4u = x5 ex , x > 0

3. (x2 + 9)u + 2xu = 0, x > 0

4. xu − 4u = 2x6 + x5 , x > 0

5. xu + u = 2x, x > 0

6. xu − u = x2 sin x, x > 0

Find the particular solution for each of the following initial value problems: 7. u − u = 2xex , u(0) = 0

π π = e2 9. (tan x)u + (sec2 x)u = 2e2x , u 4

8. xu + u = 2x, u(1) = 1 10. u − 3u = 4x3 e3x , u(0) = 1

1.2 Ordinary Diﬀerential Equations 11. (1 + x3 )u + 3x2 u = 1, u(0) = 0

9 12. u + (tan x)u = cos x, u(0) = 1

1.2.2 Second Order Linear Diﬀerential Equations As stated before, we will review some second order linear ordinary diﬀerential equations. The focus will be on second order equations, homogeneous and inhomogeneous as well. Homogeneous Equations with Constant Coeﬃcients The standard form of the second order homogeneous ordinary diﬀerential equations with constant coeﬃcients is au + bu + cu = 0, a = 0, (1.35) where a, b, and c are constants. The solution of this equation is assumed to be of the form: u(x) = erx . (1.36) Substituting this assumption into Eq. (1.35) gives the equation: erx (ar2 + br + c) = 0. (1.37) rx Since e is not zero, then we have the characteristic or the auxiliary equation: (1.38) ar2 + br + c = 0. Solving this quadratic equation leads to one of the following three cases: (i) If the roots r1 and r2 are real and r1 = r2 , then the general solution of the homogeneous equation is (1.39) u(x) = Aer1 x + Ber2 x , where A and B are constants. (ii) If the roots r1 and r2 are real and r1 = r2 = r, then the general solution of the homogeneous equation is u(x) = Aerx + Bxerx , (1.40) where A and B are constants. (iii) If the roots r1 and r2 are complex and r1 = λ + iμ, r2 = λ − iμ, then the general solution of the homogeneous equation is given by u(x) = eλx (A cos(μx) + B sin(μx)) , where A and B are constants.

(1.41)

Inhomogeneous Equations with Constant Coeﬃcients The standard form of the second order inhomogeneous ordinary diﬀerential equations with constant coeﬃcients is

10

1 Preliminaries

au + bu + cu = g(x), a = 0,

(1.42)

where a, b, and c are constants. The general solution consists of two parts, namely, complementary solution uc , and a particular solution up where the general solution is of the form: u(x) = uc (x) + up (x), where uc is the solution of the related homogeneous equation:

(1.43)

(1.44) au + bu + cu = 0, a = 0, and this is obtained as presented before. A particular solution up arises from the inhomogeneous part g(x). It is called a particular solution because it justiﬁes the inhomogeneous equation (1.42), but it is not the particular solution of the equation that is obtained from (1.43) upon using the given initial equations as will be discussed later. To obtain up (x), we use the method of undetermined coeﬃcients. To apply this method, we consider the following three types of g(x): (i) If g(x) is a polynomial given by g(x) = a0 xn + a1 xn−1 + · · · + an , then up should be assumed as up = xr (b0 xn + b1 xn−1 + · · · + bn ), r = 0, 1, 2, . . . (ii) If g(x) is an exponential function of the form: g(x) = a0 eαx ,

(1.45) (1.46) (1.47)

then up should be assumed as up = b0 xr eαx , r = 0, 1, 2, . . . (iii) If g(x) is a trigonometric function of the form:

(1.48)

g(x) = a0 sin(αx) + b0 cos(βx), then up should be assumed as

(1.49)

(1.50) up = xr (A0 sin(αx) + B0 cos(βx)), r = 0, 1, 2, . . . For other forms of g(x) such as tan x and sec x, we usually use the variation of parameters method that will not be reviewed in this text. Notice that r is the smallest nonnegative integer that will guarantee no term in up (x) is a solution of the corresponding homogeneous equation. The values of r are 0, 1 and 2. Example 1.9 Solve the following second order ODE: u − u = 0.

(1.51)

The auxiliary equation is given by r2 − 1 = 0,

(1.52)

and this gives r = ±1.

(1.53)

1.2 Ordinary Diﬀerential Equations

11

Accordingly, the general solution is given by u(x) = Aex + Be−x . It is interesting to point out that the normal form ODE:

(1.54)

u + u = 0,

(1.55)

r2 + 1 = 0,

(1.56)

r = ±i.

(1.57)

leads to the auxiliary equation: and this gives The general solution is given by u(x) = A cos x + B sin x.

(1.58)

Example 1.10 Solve the following second order ODE: u − 7u + 6u = 0.

(1.59)

The auxiliary equation is given by r2 − 7r + 6 = 0,

(1.60)

with roots given by r = 1, 6. The general solution is given by u(x) = Aex + Be6x .

(1.61) (1.62)

Example 1.11 Solve the following second order ODE: (1.63) u − 5u + 6u = 6x + 7. We ﬁrst ﬁnd uc . The auxiliary equation for the related homogeneous equation is given by r2 − 5r + 6 = 0, (1.64) with roots given by r = 2, 3. The general solution is given by u(x) = αe2x + βe3x .

(1.65) (1.66)

Noting that g(x) = 6x + 7, then a particular solution is assumed to be of the form (1.67) up = Ax + B. Since this is a particular solution, then substituting up into the inhomogeneous equation leads to 6Ax + (6B − 5A) = 6x + 7.

(1.68)

12

1 Preliminaries

Equating the coeﬃcients of like terms from both sides gives A = 1,

B = 2.

(1.69)

This in turn gives u(x) = uc + up = αe2x + βe3x + x + 2, where α and β are arbitrary constants.

(1.70)

Example 1.12 Solve the following initial value problem u + 9u = 20ex,

u(0) = 3,

y (0) = 5.

(1.71)

We ﬁrst ﬁnd uc . The auxiliary equation for the related homogeneous equation is given by (1.72) r2 + 9 = 0, where we ﬁnd r = ±3i,

i2 = −1.

(1.73)

The general solution is given by u(x) = α cos(3x) + β sin(3x). (1.74) Noting that g(x) = 20ex , then a particular solution is assumed to be of the form: up = Aex . (1.75) Since this is a particular solution, then substituting up into the inhomogeneous equation leads to 10Aex = 20ex, (1.76) so that A = 2. This in turn gives the general solution u(x) = uc + up = α cos(3x) + β sin(3x) + 2ex .

(1.77) (1.78)

Since the initial conditions are given, the numerical values for α and β should be determined. Substituting the initial values into the general solution we ﬁnd α + 2 = 3,

3β + 2 = 5,

(1.79)

β = 1.

(1.80)

where we ﬁnd α = 1,

Accordingly, the particular solution is given by u(x) = cos(3x) + sin(3x) + 2ex .

Exercises 1.2.2 Find the general solution for the following second order ODEs:

(1.81)

1.2 Ordinary Diﬀerential Equations

13

1. u − 4u + 4u = 0

2. u − 2u − 3u = 0

3. u − u − 2u = 0

4. u − 2u = 0

5. u − 6u + 9u = 0

6. u + 4u = 0

Find the general solution for the following initial value problems: 7. u − 2u + 2u = 0, u(0) = 1, u (0) = 1

8. u − 6u + 9u = 0, u(0) = 1, u (0) = 4

9. u − 3u − 10u = 0, u(0) = 2, u (0) = 3 10. u + 9u = 0, u(0) = 1, u (0) = 0 11. u − 9u = 0, u(0) = 3, u (0) = 9

12. u − 9u = 0, u(0) = 1, u (0) = 0

Use the method of undetermined coeﬃcients to ﬁnd the general solution for the following second order ODEs: 13. u − u = 1

14. u + u = 3

15. u − u = 3x

16. u − u = 2 cos x

Use the method of undetermined coeﬃcients to ﬁnd the particular solution for the following initial value problems: 17. u − u = 6, u(0) = 3, u (0) = 2

18. u + u = 6ex , u(0) = 3, u (0) = 2

19. u − u = −2 sin x, u(0) = 1, u (0) = 2 20. u − 5u + 4u = −1 + 4x, u(0) = 3, u (0) = 9

1.2.3 The Series Solution Method For diﬀerential equations of any order, with constant coeﬃcients or with variable coeﬃcients, with x = 0 is an ordinary point, we can use the series solution method to determine the series solution of the diﬀerential equation. The obtained series solution may converge the exact solution if such a closed form solution exists. If an exact solution is not obtainable, we may use a truncated number of terms of the obtained series for numerical purposes. Although the series solution can be used for equations with constant coeﬃcients or with variable coeﬃcients, where x = 0 is an ordinary point, but this method is commonly used for ordinary diﬀerential equations with variable coeﬃcients where x = 0 is an ordinary point. The series solution method assumes that the solution near an ordinary point x = 0 is given by ∞ an xn , (1.82) u(x) = n=0

or by using few terms of the series u(x) = a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + · · ·

(1.83)

Diﬀerentiating term by term gives u (x) = a1 + 2a2 x + 3a3 x2 + 4a4 x3 + 5a5 x4 + 6a6 x5 + · · · u (x) = 2a2 + 6a3 x + 12a4 x2 + 20a5 x3 + 30a6 x4 + · · · u (x) = 6a3 + 24a4 x + 60a5 x2 + 120a6 x3 + · · ·

(1.84)

14

1 Preliminaries

and so on. Substituting u(x) and its derivatives in the given diﬀerential equation, and equating coeﬃcients of like powers of x gives a recurrence relation that can be solved to determine the coeﬃcients an , n 0. Substituting the obtained values of an , n 0 in the series assumption (1.82) gives the series solution. As stated before, the series may converge to the exact solution. Otherwise, the obtained series can be truncated to any ﬁnite number of terms to be used for numerical calculations. The more terms we use will enhance the level of accuracy of the numerical approximation. It is interesting to point out that the series solution method can be used for homogeneous and inhomogeneous equations as well when x = 0 is an ordinary point. However, if x = 0 is a regular singular point of an ODE, then solution can be obtained by Frobenius method that will not be reviewed in this text. Moreover, the Taylor series of any elementary function involved in the diﬀerential equation should be used for equating the coeﬃcients. The series solution method will be illustrated by examining the following ordinary diﬀerential equations where x = 0 is an ordinary point. Some examples will give exact solutions, whereas others will provide series solutions that can be used for numerical purposes. Example 1.13 Find a series solution for the following second order ODE: u + u = 0.

(1.85)

Substituting the series assumption for u(x) and u (x) gives 2a2 + 6a3 x + 12a4 x2 + 20a5 x3 + 30a6 x4 + · · · +a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + · · · = 0, that can be rewritten by (a0 + 2a2 ) + (a1 + 6a3 )x + (a2 + 12a4 )x2 + (a3 + 20a5 )x3 +(a4 + 30a6 )x4 + · · · = 0.

(1.86)

(1.87)

This equation is satisﬁed only if the coeﬃcient of each power of x vanishes. This in turn gives the recurrence relation a0 + 2a2 = 0, a1 + 6a3 = 0, a2 + 12a4 = 0, .. .

a3 + 20a5 = 0,

(1.88)

By solving this recurrence relation, we obtain 1 1 a2 = − a0 , a3 = − a1 , 2! 3! (1.89) 1 1 1 1 a4 = − a2 = a0 , a5 = − a3 = a1 , · · · 12 4! 20 5! The solution in a series form is given by 1 2 1 3 1 4 1 5 u(x) = a0 1 − x + x + · · · + a1 x − x + x + · · · , (1.90) 2! 4! 3! 5!

1.2 Ordinary Diﬀerential Equations

15

and in closed form by (1.91) u(x) = a0 cos x + a1 sin x, where a0 and a1 are constants that will be determined for particular solution if initial conditions are given. Example 1.14 Find a series solution for the following second order ODE: u − xu − u = 0.

(1.92)

Substituting the series assumption for u(x), u (x) and u (x) gives 2a2 + 6a3 x + 12a4 x2 + 20a5 x3 + 30a6 x4 + 42a7 x5 + · · · −a1 x − 2a2 x2 − 3a3 x3 − 4a4 x4 − 5a5 x5 − · · · (1.93) −a0 − a1 x − a2 x2 − a3 x3 − a4 x4 − a5 x5 − · · · = 0, that can be rewritten by (−a0 + 2a2 ) + (−2a1 + 6a3 )x + (−3a2 + 12a4 )x2 + (−4a3 + 20a5 )x3 +(30a6 − 5a4 )x4 + (42a7 − 6a5 )x5 + · · · = 0.

(1.94)

This equation is satisﬁed only if the coeﬃcient of each power of x is zero. This gives the recurrence relation −a0 + 2a2 = 0, −2a1 + 6a3 = 0, −3a2 + 12a4 = 0, (1.95) −4a3 + 20a5 = 0, −5a4 + 30a6 = 0, −6a5 + 42a7 = 0, · · · where by solving this recurrence relation we ﬁnd 1 1 1 1 a3 = a1 , a4 = − a2 = a0 , a2 = a0 , 2 3 4 8 (1.96) 1 1 1 1 1 1 a5 = a3 = a1 , a6 = a4 = a0 , a7 = a5 = a1 , · · · 5 15 6 48 7 105 The solution in a series form is given by 1 1 1 u(x) = a0 1 − x2 + x4 + x6 + · · · 2 8 48 1 1 1 7 x + ··· , +a1 x + x3 + x5 + (1.97) 3 15 105 where a0 and a1 are constants, where a0 = u(0) and a1 = u (0). It is clear that a closed form solution is not obtainable. If a particular solution is required, then initial conditions u(0) and u (0) should be speciﬁed to determine the coeﬃcients a0 and a1 . Example 1.15 Find a series solution for the following second order ODE: u − xu + u = −x cos x.

(1.98)

Substituting the series assumption for u(x), u , and u (x), and using the Taylor series for cos x, we ﬁnd

16

1 Preliminaries

2a2 + 6a3 x + 12a4 x2 + 20a5 x3 + 30a6 x4 + 42a7 x5 + · · · −a1 x − 2a2 x2 − 3a3 x3 − 4a4 x4 − 5a5 x5 − · · · +a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + · · · 1 1 = −x + x3 − x5 + · · · , 2! 4! that can be rewritten by

(1.99)

(a0 + 2a2 ) + 6a3 x + (−a2 + 12a4 )x2 + (−2a3 + 20a5 )x3 1 1 +(30a6 − 3a4 )x4 + (42a7 − 4a5 )x5 + · · · = −x + x3 − x5 + · · · .(1.100) 2! 4! Equating the coeﬃcient of each power of x be zero and solving the recurrence relation we obtain 1 1 a2 = − a0 , a3 = − , 2 3! 1 1 1 1 1 (1.101) a3 + = , a4 = − a2 = − a0 , a5 = 12 24 10 40 120 1 1 2 1 1 a6 = a7 = a4 = − a0 , a5 − =− ,··· 10 240 21 1008 5040 The solution in a series form is given by 1 1 1 6 x + ··· u(x) = a0 1 − x2 − x4 − (1.102) 2 24 240 1 1 1 +a1 x + − x3 + x5 − x7 + · · · , 3! 5! 7! that can be rewritten as 1 1 6 1 x + ··· u(x) = a0 1 − x2 − x4 − (1.103) 2 24 240 1 1 1 +Bx + x − x3 + x5 − x7 + · · · , 3! 5! 7! by setting B = a1 − 1. Consequently, the solution is given by 1 1 6 1 (1.104) u(x) = a0 1 − x2 + x4 − x + · · · + Bx + sin x. 2 24 240 Notice that sin x gives the particular solution that arises as an eﬀect of the inhomogeneous part.

Exercises 1.2.3 Find the series solution for the following homogeneous second order ODEs: 1. u + xu + u = 0

2. u − xu + xu = 0

3. u − (1 + x)u + u = 0

4. u − u + xu = 0

Find the series solution for the following inhomogeneous second order ODEs: 5. u − u + xu = sin x

6. u − xu + xu = ex

7. u − xu = cos x

8. u − x2 u = ln(1 − x)

1.3 Leibnitz Rule for Diﬀerentiation of Integrals

17

1.3 Leibnitz Rule for Diﬀerentiation of Integrals One of the methods that will be used to solve integral equations is the conversion of the integral equation to an equivalent diﬀerential equation. The conversion is achieved by using the well-known Leibnitz rule [4,6,7] for diﬀerentiation of integrals. Let f (x, t) be continuous and ∂f ∂t be continuous in a domain of the x-t plane that includes the rectangle a x b, t0 t t1 , and let h(x) F (x) = f (x, t)dt, (1.105) g(x)

then diﬀerentiation of the integral in (1.105) exists and is given by h(x) dh(x) dg(x) dF ∂f (x, t) = f (x, h(x)) − f (x, g(x)) + dt. F (x) = dx dx dx ∂x g(x) (1.106) If g(x) = a and h(x) = b where a and b are constants, then the Leibnitz rule (1.106) reduces to b dF ∂f (x, t) F (x) = = dt, (1.107) dx ∂x a which means that diﬀerentiation and integration can be interchanged such as b b d ext dt = text dt. (1.108) dx a a It is interested to notice that Leibnitz rule is not applicable for the Abel’s singular integral equation: x u(t) F (x) = dt, 0 < α < 1. (1.109) (x − t)α 0 The integrand in this equation does not satisfy the conditions that f (x, t) be continuous, because it is unbounded at x = t. We be continuous and ∂f ∂t illustrate the Leibnitz rule by the following examples. Example 1.16 Find F (x) for the following:

cos x

F (x) =

1 + t3 dt.

(1.110)

sin x

We can set g(x) = sin x and h(x) = cos x. It is also clear that f (x, t) is a function of t only. Using Leibnitz rule (1.106) we ﬁnd that F (x) = − sin x 1 + cos3 x − cos x 1 + sin3 x. (1.111) Example 1.17 Find F (x) for the following:

x2

(x − t) cos tdt.

F (x) = x

(1.112)

18

1 Preliminaries

We can set g(x) = x, h(x) = x2 , and f (x, t) = (x − t) cos t is a function of x and t. Using Leibnitz rule (1.106) we ﬁnd that x2 2 2 F (x) = 2x(x − x ) cos x + cos tdt, (1.113) x

or equivalently F (x) = 2x(x − x2 ) cos x2 + sin x2 − sin x.

(1.114)

Remarks In this text of integral equations, we will concern ourselves in diﬀerentiation of integrals of the form: x K(x, t)u(t)dt. (1.115) F (x) = 0

In this case, Leibnitz rule (1.106) reduces to the form: x ∂K(x, t) u(t)dt. F (x) = K(x, x)u(x) + ∂x 0 This will be illustrated by the following examples.

(1.116)

Example 1.18 Find F (x) for the following:

x

F (x) = 0

(x − t)u(t)dt.

Applying the reduced Leibnitz rule (1.116) yields x F (x) = u(t)dt.

(1.117)

(1.118)

0

Example 1.19 Find F (x) and F (x) for the following: x xtu(t)dt. F (x) =

(1.119)

0

Applying the reduced Leibnitz rule (1.116) gives x F (x) = x2 u(x) + tu(t)dt, 0

(1.120)

F (x) = x2 u (x) + 3xu(x). Example 1.20 Find F (x), F (x), F (x) and F (iv) (x) for the following integral x (x − t)3 u(t)dt. F (x) = 0

Applying the reduced Leibnitz rule (1.116) four times gives

(1.121)

1.4 Reducing Multiple Integrals to Single Integrals

F (x) = F (x) =

x 0

3(x − t)2 u(t)dt,

x

6u(t)dt,

F (x) =

19

x

6(x − t)u(t)dt,

0

(1.122)

F (iv) (x) = 6u(x).

0

Exercises 1.3 Find F (x) for the following integrals: x 2 2 1. F (x) = e−x t dt 3. F (x) =

0 x 0

sin(x2 + t2 )dt

Find F (x) for the following integrals: x (x − t)u(t)dt 5. F (x) = 0 x

7. F (x) =

0

(x − t)3 u(t)dt

2. F (x) =

x2

ln(1 + xt)dt

x x

4. F (x) =

6. F (x) =

cosh(x3 + t3 )dt

0 x

0 x

8. F (x) =

0

(x − t)2 u(t)dt (x − t)4 u(t)dt

Diﬀerentiate both sides of the following equations: x x 1 9. x3 + x6 = (4 + x − t)u(t)dt 10. 1 + xex = ex−t u(t)dt 6 0 0 x 11. 2x2 + 3x3 = (6 + 5x − 5t)u(t)dt 0

12. sinh x + ln(sin x) =

x 0

(3 + x − t)u(t)dt, 0 < x <

π 2

Diﬀerentiate the following F (x) as many times as you need to get rid of the integral sign: x x 13. F (x) = x + (x − t)u(t)dt 14. F (x) = x2 + (x − t)2 u(t)dt 15. F (x) = 1 +

0 x

0

3

(x − t) u(t)dt

x

16. F (x) = e +

0 x

0

(x − t)4 u(t)dt

Use Leibnitz rule to prove the following identities: x (x − t)n u(t)dt, show that F (n+1) = n!u(x), n 0. 17. If F (x) = 18. F (x) =

0

x

0

tn (x − t)m dt, show that F (m) =

m! n+1 x , n+1

m and n are positive integers.

1.4 Reducing Multiple Integrals to Single Integrals It will be seen later that we can convert initial value problems and other problems to integral equations. It is normal to outline the formula that will

20

1 Preliminaries

reduce multiple integrals to single integrals. We will ﬁrst show that the double integral can be reduced to a single integral by using the formula: x x1 x F (t)dtdx1 = (x − t)F (t)dt. (1.123) 0

0

0

This can be easily proved by two ways. The ﬁrst way is to set x (x − t)F (t)dt, G(x) =

(1.124)

where G(0) = 0. Diﬀerentiating both sides of (1.124) gives x G (x) = F (t)dt,

(1.125)

0

0

obtained by using the reduced Leibnitz rule. Now by integrating both sides of the last equation from 0 to x yields x x1 G(x) = F (t)dtdx1 . (1.126) 0

0

Consequently, the right side of the two equations (1.124) and (1.126) are equivalent. This completes the proof. For the second method we will use the concept of integration by parts. Recall that udv = uv − vdu, x1 (1.127) F (t)dt, u(x1 ) = 0

then we ﬁnd x x1 0

0

x x F (t)dtdx1 = x1 F (t)dt − x1 F (x1 )dx1 0 x0 x 0 =x F (t)dt − x1 F (x1 )dx1 0 0 x = (x − t)F (t)dt,

x1

(1.128)

0

obtained upon setting x1 = t. The general formula that converts multiple integrals to a single integral is given by xn−1 x x1 x 1 ··· u(xn )dxn dxn−1 · · · dx1 = (x − t)n−1 u(t)dt. (n − 1)! 0 0 0 0 (1.129) The conversion formula (1.129) is very useful and facilitates the calculation works. Moreover, this formula will be used to convert initial value problems to Volterra integral equations. Corollary As a result to (1.129) we can easily show the following corollary

1.4 Reducing Multiple Integrals to Single Integrals

0

x x 0

···

x

1 (x − t)u(t)dtdt · · · dt = n! 0 n integrals

21

0

x

(x − t)n u(t)dt

(1.130)

Example 1.21 Convert the following multiple integral to a single integral: x x1 u(x2 )dx1 dx.

(1.131)

Using the formula (1.129) we obtain x x1 x u(x2 )dx1 dx = (x − t)u(t)dt.

(1.132)

0

0

0

0

0

Example 1.22 Convert the following multiple integral to a single integral: x x1 x2 u(x3 )dx2 dx1 dx. 0

0

(1.133)

0

Using the formula (1.129) we obtain x x1 x2 1 x u(x3 )dx2 dx1 dx = (x − t)2 u(t)dt. 2! 0 0 0 0

(1.134)

Example 1.23 Convert the following multiple integral to a single integral: x x x (x − t)dtdtdt. 0

0

(1.135)

0

Using the corollary (1.130) we obtain x x x 1 x (x − t)dtdtdt = (x − t)3 u(t)dt. 3! 0 0 0 0

(1.136)

Example 1.24 Convert the following multiple integral to a single integral: x x1 (x − t)u(x1 )dtdx1 . 0

(1.137)

0

Using the corollary (1.130) we obtain x x1 1 x (x − t)u(x1 )dtdx1 = (x − t)2 u(t)dt. 2 0 0 0

(1.138)

Example 1.25 Convert the following multiple integral to a single integral: x x1 (x − t)2 u(x1 )dtdx1 . 0

0

Using the corollary (1.130) we obtain

(1.139)

22

1 Preliminaries

0

x

x1

0

(x − t)2 u(x1 )dtdx1 =

1 3

x 0

(x − t)3 u(t)dt.

(1.140)

Exercises 1.4 Prove the following: x x1 1 x (x − t)3 u(x1 )dtdx1 = (x − t)4 u(t)dt 1. 4 0 0 0 x x 1 1 x 2. (x − t)4 u(x1 )dtdx1 = (x − t)5 u(t)dt 5 0 0 0 x x x x1 1 3. (x − t)u(x1 )dtdx1 + (x − t)2 u(x1 )dtdx1 0

0

1 = 6 x 4. 0

=

0

x

0 x1

0

1 4

0

(x − t) (3 + 2(x − t))u(t)dt

u(x1 )dtdx1 +

x

0

2

x

0

x1

0

(x − t)u(x1 )dtdx1 +

x

0

x1

0

(x − t)3 u(x1 )dtdx1

(x − t)2 4 + 2(x − t) + (x − t)3 u(t)dt

1.5 Laplace Transform In this section we will review only the basic concepts of the Laplace transform method. The details can be found in any text of ordinary diﬀerential equations. The Laplace transform method is a powerful tool used for solving diﬀerential and integral equations. The Laplace transform [1–4,7] changes differential equations and integral equations to polynomial equations that can be easily solved, and hence by using the inverse Laplace transform gives the solution of the examined equation. The Laplace transform of a function f (x), deﬁned for x 0, is deﬁned by ∞ e−sx f (x)dx, (1.141) F (s) = L{f (x)} = 0

where s is real, and L is called the Laplace transform operator. The Laplace transform F (s) may fail to exist. If f (x) has inﬁnite discontinuities or if it grows up rapidly, then F (s) does not exist. Moreover, an important necessary condition for the existence of the Laplace transform F (s) is that F (s) must vanish as s approaches inﬁnity. This means that lim F (s) = 0. (1.142) s→∞

In other words, the conditions for the existence of a Laplace transform F (s) of any function f (x) are: 1. f (x) is piecewise continuous on the interval of integration 0 x < A for any positive A,

1.5 Laplace Transform

23

2. f (x) is of exponential order eax as x → ∞, i.e. |f (x)| Keax , x M , where a is real constant, and K and M are positive constants. Accordingly, the Laplace transform F (s) exists and must satisfy lim F (s) = 0. (1.143) s→∞

1.5.1 Properties of Laplace Transforms From the deﬁnition of the Laplace transform given in (1.141), we can easily derive the following properties of the Laplace transforms: 1). Constant Multiple: L{af (x)} = aL{f (x)}, a is a constant.

(1.144)

For example: L{4ex} = 4L{ex } =

4 . s−1

(1.145)

2). Linearity Property: L{af (x) + bg(x)} = aL{f (x)} + bL{g(x)}, a, b are constants. (1.146) For example: 4 6 (1.147) L{4x + 3x2 } = 4L{x} + 3L{x2 } = 2 + 3 . s s 3). Multiplication by x: d (1.148) L{xf (x)} = − L{f (x)} = −F (s). ds For example: d 1 2s d = 2 L{x sin x} = − L{sin x} = − . (1.149) 2 ds ds s + 1 (s + 1)2 To use the Laplace transform L for solving initial value problems or integral equations, we use the following table of elementary Laplace transforms as shown below: Table 1.1 Elementary Laplace Transforms f (x) c x xn eax

F (s) = L{f (x)} =

∞ 0

e−sx f (x)dx

c , s>0 s 1 , s>0 s2 n! Γ(n + 1) = , s > 0, Re n > −1 sn+1 sn+1 1 , s>a s−a

24

1 Preliminaries f (x) sin ax cos ax sin2 ax cos2 ax x sin ax x cos ax sinh ax cosh ax sinh2 ax cosh2 ax x sinh ax x cosh ax xn eax eax sin bx eax cos bx eax sinh bx eax cosh bx H(x − a) δ(x)

F (s) = L{f (x)} =

Continued ∞ 0

e

−sx

f (x)dx

a s 2 + a2 s s 2 + a2 2a2 , Re(s) > |Im(a)| s(s2 + 4a2 ) 2 2 s + 2a , Re(s) > |Im(a)| s(s2 + 4a2 ) 2as (s2 + a2 )2 s 2 − a2 (s2 + a2 )2 a , s > |a| s 2 − a2 s , s > |a| s 2 − a2 2a2 , Re(s) > |Im(a)| s(s2 − 4a2 ) s2 − 2a2 , Re(s) > |Im(a)| s(s2 − 4a2 ) 2as , s > |a| (s2 − a2 )2 2 2 s +a , s > |a| (s2 − a2 )2 n! , s > a, n is a positive integer (s − a)n+1 b , s>a (s − a)2 + b2 s−a , s>a (s − a)2 + b2 b , s>a (s − a)2 − b2 s−a , s>a (s − a)2 − b2 s−1 e−as , a 0 1

δ(x − a)

e−as , a 0

δ (x − a)

se−as , a 0

1.5 Laplace Transform

25

4). Laplace Transforms of Derivatives: L{f (x)} = sL{f (x)} − f (0), L{f (x)} = s2 L{f (x)} − sf (0) − f (0), L{f (x)} = s3 L{f (x)} − s2 f (0) − sf (0) − f (0), .. . L{f (n) (x)} = sn L{f (x)} − sn−1 f (0) − · · · − sf (n−2) (0) − f (n−1) (0). (1.150) 5). Inverse Laplace Transform If the Laplace transform of f (x) is F (s), then we say that the inverse Laplace transform of F (s) is f (x). In other words, we write (1.151) L−1 {F (s)} = f (x), where L−1 is the operator of the inverse Laplace transform. The linearity property holds also for the inverse Laplace transform. This means that L−1 {aF (s) + bG(s)} = aL−1 {F (s)} + bL−1 {G(s)} (1.152) = af (x) + bg(x). Notice that the computer symbolic systems such as Maple and Mathematica can be used to ﬁnd the Laplace transform and the inverse Laplace transform. The Laplace transform method and the inverse Laplace transform method will be illustrated by using the following examples. Example 1.26 Solve the following initial value problem: y + y = 0, y(0) = 1,

y (0) = 1.

(1.153)

By taking Laplace transforms of both sides of the ODE, we use L{y(x)} = Y (s), L{y (x)} = s2 L{y(x)} − sy(0) − y (0)

(1.154)

= s2 Y (s) − s − 1, obtained upon using the initial conditions. Substituting this into the ODE gives s 1 Y (s) = 2 + . (1.155) s + 1 s2 + 1 To determine the solution y(x) we then take the inverse Laplace transform L−1 to both sides of the last equation to ﬁnd s 1 −1 L−1 {Y (s)} = L−1 + L . (1.156) s2 + 1 s2 + 1 This in turn gives the solution by y(x) = cos x + sin x,

(1.157)

26

1 Preliminaries

obtained upon using the table of Laplace transforms. Notice that we obtained the particular solution for the diﬀerential equation. This is due to the fact that the initial conditions were used in the solution. Example 1.27 Solve the following initial value problem y − y = 0, y(0) = 2, y (0) = 1. By taking Laplace transforms of both sides of the ODE, we set L{y(x)} = Y (s), L{y (x)} = sY (s) − 2, L{y (x)} = s2 L{y(x)} − sy(0) − y (0)

(1.158)

(1.159)

= s2 Y (s) − 2s − 1, obtained by using the initial conditions. Substituting this into the ODE gives 1 1 + . (1.160) Y (s) = s−1 s To determine the solution y(x) we then the take inverse Laplace transform L−1 of both sides of the last equation to ﬁnd 1 1 + L−1 . (1.161) L−1 {Y (s)} = L−1 s−1 s This in turn gives the solution by (1.162) y(x) = ex + 1, obtained upon using the table of Laplace transforms. 6). The Convolution Theorem for Laplace Transform This is an important theorem that will be used in solving integral equations. The kernel K(x, t) of the integral equation: h(x) u(x) = f (x) + λ K(x, t)u(t)dt, (1.163) g(x)

is termed diﬀerence kernel if it depends on the diﬀerence x − t. Examples of the diﬀerence kernels are ex−t, sin(x − t), and cosh(x − t). The integral equation (1.163) can be expressed as h(x) K(x − t)u(t)dt. (1.164) u(x) = f (x) + λ g(x)

Consider two functions f1 (x) and f2 (x) that possess the conditions needed for the existence of Laplace transform for each. Let the Laplace transforms for the functions f1 (x) and f2 (x) be given by L{f1 (x)} = F1 (s), (1.165) L{f2 (x)} = F2 (s). The Laplace convolution product of these two functions is deﬁned by

1.5 Laplace Transform

27

(f1 ∗ f2 )(x) =

0

or

x

(f2 ∗ f1 )(x) =

0

f1 (x − t)f2 (t)dt,

(1.166)

f2 (x − t)f1 (t)dt.

(1.167)

x

Recall that (f1 ∗ f2 )(x) = (f2 ∗ f1 )(x).

(1.168)

We can easily show that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x f1 (x − t)f2 (t)dt = F1 (s)F2 (s). (1.169) L{(f1 ∗ f2 )(x)} = L 0

It was stated before, that this theorem will be used in the coming chapters. To illustrate the use of this theorem we examine the following example. Example 1.28 Find the Laplace transform of

x+ 0

x

(x − t)y(t)dt.

(1.170)

Notice that the kernel depends on the diﬀerence x − t. The integral includes f1 (x) = x and f2 (x) = y(x). The integral is the convolution product (f1 ∗ f2 )(x). This means that if we take Laplace transform of each term we obtain x (x − t)y(t)dt = L{x} + L{x}L{y(t)}. (1.171) L{x} + L 0

Using the table of Laplace transforms gives 1 1 + 2 Y (s). s2 s

(1.172)

Example 1.29 Solve the following integral equation x x xe = ex−t y(t)dt.

(1.173)

0

Notice that f1 (x) = ex and f2 (x) = y(x). The right hand side is the convolution product (f1 ∗ f2 )(x). This means that if we take Laplace transforms of both sides we obtain x

L{xex } = L

ex−t y(t)dt 0

= L{ex }L{y(t)}.

Using the table of Laplace transforms gives 1 1 = Y (s), (s − 1)2 s−1 that gives 1 . Y (s) = s−1

(1.174)

(1.175)

(1.176)

28

1 Preliminaries

From this we ﬁnd the solution is

1 (1.177) = ex . s−1 The Laplace transform method will be used for solving Volterra integral equations of the ﬁrst and the second type in coming chapters. y(x) = L−1

Exercises 1.5 Solve the given ODEs: 1. y + 4y = 0, y(0) = 0, y (0) = 1

2. y + 4y = 4x, y(0) = 0, y (0) = 1

3. y − y = −2x, y(0) = 0, y (0) = 1

4. y − 3y + 2y = 0, y(0) = 2, y (0) = 3

Find the Laplace transform of the following expressions: 5. x + sin x

6. ex − cos x

7. 1 + xex

8. sin x + sinh x

Find the Laplace transform of the following expressions that include convolution products: x x 9. sinh(x − t)y(t)dt 10. x2 + ex−t y(t)dt 0

0

x

11. 0

(x − t)ex−t y(t)dt

12. 1 + x −

x 0

(x − t)y(t)dt

Find the inverse Laplace transform of the following: 4 1 + 3 s2 + 1 s 1 s 15. F (s) = 2 + 2 s +1 s −4 13. F (s) =

3 1 + 2 s2 − 1 s +9 1 1 16. F (s) = − s(s2 + 1) s(s2 − 1) 14. F (s) =

1.6 Inﬁnite Geometric Series A geometric sequence is a sequence of numbers where each term after the ﬁrst is obtained by multiplying the previous term by a non-zero number r called the common ratio. In other words, a sequence is geometric if there is a ﬁxed non-zero number r such that an+1 = an r, n 1. (1.178) This means that the geometric sequence can be written in a general form as (1.179) a1 , a1 r, a1 r2 , . . . , a1 rn−1 , . . . , where a1 is the starting value of the sequence and r is the common ratio. The associated geometric series is obtained as the sum of the terms of the geometric series, and therefore given by n Sn = a1 rk = a1 + a1 r + a1 r2 + a1 r3 + a1 r4 + · · · + a1 rn . (1.180) k=0

1.6 Inﬁnite Geometric Series

29

The sum of the ﬁrst n terms of a geometric sequence is given by a1 (1 − rn ) , r = 1. (1.181) Sn = 1−r An inﬁnite geometric series converges if and only if |r| < 1. Otherwise it diverges. The sum of inﬁnite geometric series, for |r| < 1, is given by a1 lim Sn = , (1.182) n→∞ 1−r obtained from (1.181) by noting that lim rn = 0,

n→∞

|r| < 1.

(1.183)

Some of the methods that will be used in this text may give the solutions in an inﬁnite series. Some of the obtained series include inﬁnite geometric series. For this reason we will study examples of inﬁnite geometric series. Example 1.30 Find the sum of the inﬁnite geometric series 8 2 4 + ··· (1.184) 1+ + + 3 9 27 The ﬁrst value of the sequence and the common ratio are given by a1 = 1 and r = 23 respectively. The sum is therefore given by 1 S= = 3. (1.185) 1 − 23 Example 1.31 Find the sum of the inﬁnite geometric series: e−1 + e−2 + e−3 + e−4 + · · · (1.186) −1 −1 The ﬁrst term and the common ratio are a1 = e and r = e < 1. The sum is therefore given by e−1 1 . (1.187) S= = −1 1−e e−1 Example 1.32 Find the sum of the inﬁnite geometric series: n n2 n3 x+ x+ x+ x + · · · , 0 < n < 4. (1.188) 4 16 64 n It is obvious that a1 = x and r = 4 < 1. Consequently, the sum of this inﬁnite series is given by x 4 (1.189) S= n = 4 − n x, 0 < n < 4. 1− 4 Example 1.33 Find the sum of the inﬁnite geometric series:

30

1 Preliminaries

1 1 1 + + 3 + · · · , x > 1. (1.190) x x2 x 1 1 It is obvious that a1 = x and r = x < 1. The sum is therefore given by S=

1 x

1−

1 x

=

1 , x−1

x > 1.

(1.191)

Example 1.34 Find the sum of the inﬁnite geometric series: 1 1 1 1 − + − + ··· (1.192) 2 4 8 It is obvious that a1 = 1 and r = − 21 , |r| < 1. The sum is therefore given by 1 2 S= (1.193) = . 3 1 + 12 Exercises 1.6 Find the sum of the following inﬁnite series: 1 1 1 1 1 1 1 1 + + + + ··· 2. − + − + ··· 2 4 8 16 2 4 8 16 5 5 5 5 3. x + x+ x+ x + ··· 6 36 216 1296 6 6 6 6 4. sin x + sin x + sin x + sin x + · · · 7 49 343 2401 n n2 n3 6. x + x + x+ x + ··· , 0 < n < 9 5. e−2 + e−4 + e−6 + e−8 + · · · 9 81 729 π π e e2 π e3 e4 8. + 2 + 3 + 4 + ··· 7. πx + x + x + x + · · · 2 4 8 π π π π ln x ln x 2 d ln x + + + ··· = , x > 0 9. Show that dx 2 4 x e+1 1 1 1 + 2 + 3 + · · · dx = 1, x > 1 10. Show that 2 x x x 1.

References 1. C. Baker, The Numerical Treatment of Integral Equations, Oxford University Press, London, (1977). 2. M. Bocher, Integral Equations,Cambridge University Press, London, (1974). 3. L.G. Chambers, Integral Equations, A Short Course, International Textbook Company, London, (1976). 4. A.J. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 5. R.P. Kanwal and K.C. Liu, A Taylor expansion approach for solving integral equations, Int. J. Math. Educ. Sci. Technol., 20(3) (1989) 411–414.

References

31

6. R. Kress, Linear Integral Equations, Springer, Berlin, (1999). 7. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).

Chapter 2

Introductory Concepts of Integral Equations

As stated in the previous chapter, an integral equation is the equation in which the unknown function u(x) appears inside an integral sign [1–5]. The most standard type of integral equation in u(x) is of the form h(x) u(x) = f (x) + λ K(x, t)u(t)dt, (2.1) g(x)

where g(x) and h(x) are the limits of integration, λ is a constant parameter, and K(x, t) is a known function, of two variables x and t, called the kernel or the nucleus of the integral equation. The unknown function u(x) that will be determined appears inside the integral sign. In many other cases, the unknown function u(x) appears inside and outside the integral sign. The functions f (x) and K(x, t) are given in advance. It is to be noted that the limits of integration g(x) and h(x) may be both variables, constants, or mixed. Integral equations appear in many forms. Two distinct ways that depend on the limits of integration are used to characterize integral equations, namely: 1. If the limits of integration are ﬁxed, the integral equation is called a Fredholm integral equation given in the form: b u(x) = f (x) + λ K(x, t)u(t)dt, (2.2) a

where a and b are constants. 2. If at least one limit is a variable, the equation is called a Volterra integral equation given in the form: x u(x) = f (x) + λ K(x, t)u(t)dt. (2.3) a

Moreover, two other distinct kinds, that depend on the appearance of the unknown function u(x), are deﬁned as follows: 1. If the unknown function u(x) appears only under the integral sign of Fredholm or Volterra equation, the integral equation is called a ﬁrst kind Fredholm or Volterra integral equation respectively. A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

34

2 Introductory Concepts of Integral Equations

2. If the unknown function u(x) appears both inside and outside the integral sign of Fredholm or Volterra equation, the integral equation is called a second kind Fredholm or Volterra equation integral equation respectively. In all Fredholm or Volterra integral equations presented above, if f (x) is identically zero, the resulting equation: b u(x) = λ K(x, t)u(t)dt (2.4)

or

a x

K(x, t)u(t)dt

u(x) = λ

(2.5)

a

is called homogeneous Fredholm or homogeneous Volterra integral equation respectively. It is interesting to point out that any equation that includes both integrals and derivatives of the unknown function u(x) is called integro-diﬀerential equation. The Fredholm integro-diﬀerential equation is of the form: b dk u u(k) (x) = f (x) + λ K(x, t)u(t)dt, u(k) = k . (2.6) dx a However, the Volterra integro-diﬀerential equation is of the form: x dk u K(x, t)u(t)dt, u(k) = k . (2.7) u(k) (x) = f (x) + λ dx a The integro-diﬀerential equations [6] will be deﬁned and classiﬁed in this text.

2.1 Classiﬁcation of Integral Equations Integral equations appear in many types. The types depend mainly on the limits of integration and the kernel of the equation. In this text we will be concerned on the following types of integral equations.

2.1.1 Fredholm Integral Equations For Fredholm integral equations, the limits of integration are ﬁxed. Moreover, the unknown function u(x) may appear only inside integral equation in the form: b

f (x) =

K(x, t)u(t)dt.

(2.8)

a

This is called Fredholm integral equation of the ﬁrst kind. However, for Fredholm integral equations of the second kind, the unknown function u(x) appears inside and outside the integral sign. The second kind is represented by the form: b

K(x, t)u(t)dt.

u(x) = f (x) + λ a

(2.9)

2.1 Classiﬁcation of Integral Equations

35

Examples of the two kinds are given by 1 sin x − x cos x = sin(xt)u(t)dt, x2 0 and 1 1 u(x) = x + (x − t)u(t)dt, 2 −1

(2.10)

(2.11)

respectively.

2.1.2 Volterra Integral Equations In Volterra integral equations, at least one of the limits of integration is a variable. For the ﬁrst kind Volterra integral equations, the unknown function u(x) appears only inside integral sign in the form: x f (x) = K(x, t)u(t)dt. (2.12) 0

However, Volterra integral equations of the second kind, the unknown function u(x) appears inside and outside the integral sign. The second kind is represented by the form: x K(x, t)u(t)dt. (2.13) u(x) = f (x) + λ 0

Examples of the Volterra integral equations of the ﬁrst kind are x xe−x = et−x u(t)dt,

(2.14)

0

and 5x2 + x3 =

x

0

(5 + 3x − 3t)u(t)dt.

(2.15)

However, examples of the Volterra integral equations of the second kind are x u(x) = 1 − u(t)dt, (2.16) 0

and u(x) = x +

0

x

(x − t)u(t)dt.

(2.17)

2.1.3 Volterra-Fredholm Integral Equations The Volterra-Fredholm integral equations [6,7] arise from parabolic boundary value problems, from the mathematical modelling of the spatio-temporal development of an epidemic, and from various physical and biological models. The Volterra-Fredholm integral equations appear in the literature in two

36

2 Introductory Concepts of Integral Equations

forms, namely

x

u(x) = f (x) + λ1 a

and

b

K1 (x, t)u(t)dt + λ2

K2 (x, t)u(t)dt,

(2.18)

a

t

u(x, t) = f (x, t) + λ 0

Ω

F (x, t, ξ, τ, u(ξ, τ ))dξdτ, (x, t) ∈ Ω × [0, T ], (2.19)

where f (x, t) and F (x, t, ξ, τ, u(ξ, τ )) are analytic functions on D = Ω×[0, T ], and Ω is a closed subset of Rn , n = 1, 2, 3. It is interesting to note that (2.18) contains disjoint Volterra and Fredholm integral equations, whereas (2.19) contains mixed Volterra and Fredholm integral equations. Moreover, the unknown functions u(x) and u(x, t) appear inside and outside the integral signs. This is a characteristic feature of a second kind integral equation. If the unknown functions appear only inside the integral signs, the resulting equations are of ﬁrst kind, but will not be examined in this text. Examples of the two types are given by x 1 2 xu(t)dt − tu(t)dt, (2.20) u(x) = 6x + 3x + 2 − 0

0

and 1 1 u(x, t) = x + t3 + t2 − t − 2 2

t 0

1 0

(τ − ξ)dξdτ.

(2.21)

2.1.4 Singular Integral Equations Volterra integral equations of the ﬁrst kind [4,7] h(x) K(x, t)u(t)dt f (x) = λ

(2.22)

g(x)

or of the second kind

h(x)

K(x, t)u(t)dt

u(x) = f (x) +

(2.23)

g(x)

are called singular if one of the limits of integration g(x), h(x) or both are inﬁnite. Moreover, the previous two equations are called singular if the kernel K(x, t) becomes unbounded at one or more points in the interval of integration. In this text we will focus our concern on equations of the form: x 1 f (x) = u(t)dt, 0 < α < 1, (2.24) (x − t)α 0 or of the second kind: x 1 u(t)dt, 0 < α < 1. (2.25) u(x) = f (x) + α 0 (x − t) The last two standard forms are called generalized Abel’s integral equation and weakly singular integral equations respectively. For α = 12 , the equation:

2.2 Classiﬁcation of Integro-Diﬀerential Equations

37

x

1 u(t)dt (2.26) x−t 0 is called the Abel’s singular integral equation. It is to be noted that the kernel in each equation becomes inﬁnity at the upper limit t = x. Examples of Abel’s integral equation, generalized Abel’s integral equation, and the weakly singular integral equation are given by x √ 1 √ u(t)dt, (2.27) x= x−t x0 1 x3 = (2.28) 1 u(t)dt, 0 (x − t) 3 √

f (x) =

and u(x) = 1 +

√

x

x+ 0

1 1

(x − t) 3

u(t)dt,

(2.29)

respectively. Exercises 2.1 For each of the following integral equations, classify as Fredholm, Volterra, or Volterra-Fredholm integral equation and ﬁnd its kind. Classify the equation as singular or not. x x u(t)dt 2. x = (1 + x − t)u(t)dt 1. u(x) = 1 + 0

0

x

3. u(x) = e + e − 1 −

1 0

u(t)dt

π 4. x + 1 − = 2

π 2 (x − t)u(t)dt 0

x 1 1 3 1 (x − t)u(t)dt 6. u(x) = x + x3 − (x − t)u(t)dt 5. u(x) = x − − 2 3 6 0 0 x 1 1 2 1 1 (x − t)u(t)dt 8. x2 − x + = (x − t)u(t)dt 7. x3 = 6 2 3 4 0 0 x 1 3 1 9. u(x) = x + x3 − (x − t)u(t)dt − xu(t)dt 2 6 0 0 t 1 1 1 (τ − ξ)dξdτ 10. u(x, t) = x + t3 + t2 − t − 2 2 0 0 x x √ 1 1 √ u(t)dt 11. x3 + x = 12. u(x) = 1 + x2 + 5 u(t)dt x −t 0 (x − t) 6 0

2.2 Classiﬁcation of Integro-Diﬀerential Equations Integro-diﬀerential equations appear in many scientiﬁc applications, especially when we convert initial value problems or boundary value problems to integral equations. The integro-diﬀerential equations contain both integral

38

2 Introductory Concepts of Integral Equations

and diﬀerential operators. The derivatives of the unknown functions may appear to any order. In classifying integro-diﬀerential equations, we will follow the same category used before.

2.2.1 Fredholm Integro-Diﬀerential Equations Fredholm integro-diﬀerential equations appear when we convert diﬀerential equations to integral equations. The Fredholm integro-diﬀerential equation contains the unknown function u(x) and one of its derivatives u(n) (x), n 1 inside and outside the integral sign respectively. The limits of integration in this case are ﬁxed as in the Fredholm integral equations. The equation is labeled as integro-diﬀerential because it contains diﬀerential and integral operators in the same equation. It is important to note that initial conditions should be given for Fredholm integro-diﬀerential equations to obtain the particular solutions. The Fredholm integro-diﬀerential equation appears in the form: b

u(n) (x) = f (x) + λ

K(x, t)u(t)dt,

(2.30)

a

where u(n) indicates the nth derivative of u(x). Other derivatives of less order may appear with u(n) at the left side. Examples of the Fredholm integrodiﬀerential equations are given by 1 1 u (x) = 1 − x + xu(t)dt, u(0) = 0, (2.31) 3 0 and π2 xtu(t)dt, u(0) = 0, u (0) = 1. (2.32) u (x) + u (x) = x − sin x − 0

2.2.2 Volterra Integro-Diﬀerential Equations Volterra integro-diﬀerential equations appear when we convert initial value problems to integral equations. The Volterra integro-diﬀerential equation contains the unknown function u(x) and one of its derivatives u(n) (x), n 1 inside and outside the integral sign. At least one of the limits of integration in this case is a variable as in the Volterra integral equations. The equation is called integro-diﬀerential because diﬀerential and integral operators are involved in the same equation. It is important to note that initial conditions should be given for Volterra integro-diﬀerential equations to determine the particular solutions. The Volterra integro-diﬀerential equation appears in the form: x u(n) (x) = f (x) + λ K(x, t)u(t)dt, (2.33) 0

2.2 Classiﬁcation of Integro-Diﬀerential Equations

39

where u(n) indicates the nth derivative of u(x). Other derivatives of less order may appear with u(n) at the left side. Examples of the Volterra integrodiﬀerential equations are given by x 1 2 x u (x) = −1 + x − xe − tu(t)dt, u(0) = 0, (2.34) 2 0 and x

u (x)+u (x) = 1−x(sin x+cos x)−

0

tu(t)dt, u(0) = −1, u (0) = 1. (2.35)

2.2.3 Volterra-Fredholm Integro-Diﬀerential Equations The Volterra-Fredholm integro-diﬀerential equations arise in the same manner as Volterra-Fredholm integral equations with one or more of ordinary derivatives in addition to the integral operators. The Volterra-Fredholm integro-diﬀerential equations appear in the literature in two forms, namely x b (n) K1 (x, t)u(t)dt + λ2 K2 (x, t)u(t)dt, (2.36) u (x) = f (x) + λ1 a

and u

(n)

a

t (x, t) = f (x, t) + λ 0

Ω

F (x, t, ξ, τ, u(ξ, τ ))dξdτ, (x, t) ∈ Ω × [0, T ],

(2.37) where f (x, t) and F (x, t, ξ, τ, u(ξ, τ )) are analytic functions on D = Ω×[0, T ], and Ω is a closed subset of Rn , n = 1, 2, 3. It is interesting to note that (2.36) contains disjoint Volterra and Fredholm integral equations, whereas (2.37) contains mixed integrals. Other derivatives of less order may appear as well. Moreover, the unknown functions u(x) and u(x, t) appear inside and outside the integral signs. This is a characteristic feature of a second kind integral equation. If the unknown functions appear only inside the integral signs, the resulting equations are of ﬁrst kind. Initial conditions should be given to determine the particular solution. Examples of the two types are given by x 1 u (x) = 24x + x4 + 3 − (x − t)u(t)dt − tu(t)dt, u(0) = 0, (2.38) 0

and 1 1 u (x, t) = 1 + t3 + t2 − t − 2 2

0

t 0

1 0

(τ − ξ)dξdτ,

u(0, t) = t3 .

(2.39)

Exercises 2.2 For each of the following integro-diﬀerential equations, classify as Fredholm, Volterra, or Volterra-Fredholm integro-equation

40 1. u (x) = 1 +

2 Introductory Concepts of Integral Equations

2. u (x) = x +

x

xu(t)dt, u(0) = 0

0

1

0

(1 + x − t)u(t)dt, u(0) = 1, u (0) = 0

3. u (x) + u(x) = x +

tu(t)dt +

0

4. u (x) + u (x) = x + 5. u (x) + u(x) = x + 6. u (x) = 1 +

x

0

x

1

0

0

x

0

1

tu(t)dt +

u(t)dt, u(0) = 0, u (0) = 1 1

0

u(t)dt, u(0) = 0, u (0) = 1, u (0) = 1

(x − t)u(t)dt, u(0) = 1

tu(t)dt, u(0) = 0, u (0) = 1

2.3 Linearity and Homogeneity Integral equations and integro-diﬀerential equations fall into two other types of classiﬁcations according to linearity and homogeneity concepts. These two concepts play a major role in the structure of the solutions. In what follows we highlight the deﬁnitions of these concepts.

2.3.1 Linearity Concept If the exponent of the unknown function u(x) inside the integral sign is one, the integral equation or the integro-diﬀerential equation is called linear [6]. If the unknown function u(x) has exponent other than one, or if the equation contains nonlinear functions of u(x), such as eu , sinh u, cos u, ln(1 + u), the integral equation or the integro-diﬀerential equation is called nonlinear. To explain this concept, we consider the equations: x (x − t)u(t)dt, (2.40) u(x) = 1 − 0

u(x) = 1 − u(x) = 1 + 0

and

1

u (x) = 1 + 0

(x − t)u(t)dt,

(2.41)

(1 + x − t)u4 (t)dt,

(2.42)

0 x

1

xteu(t) dt,

u(0) = 1.

(2.43)

2.3 Linearity and Homogeneity

41

The ﬁrst two examples are linear Volterra and Fredholm integral equations respectively, whereas the last two are nonlinear Volterra integral equation and nonlinear Fredholm integro-diﬀerential equation respectively. It is important to point out that linear equations, except Fredholm integral equations of the ﬁrst kind, give a unique solution if such a solution exists. However, solution of nonlinear equation may not be unique. Nonlinear equations usually give more than one solution and it is not usually easy to handle. Both linear and nonlinear integral equations of any kind will be investigated in this text by using traditional and new methods.

2.3.2 Homogeneity Concept Integral equations and integro-diﬀerential equations of the second kind are classiﬁed as homogeneous or inhomogeneous, if the function f (x) in the second kind of Volterra or Fredholm integral equations or integro-diﬀerential equations is identically zero, the equation is called homogeneous. Otherwise it is called inhomogeneous. Notice that this property holds for equations of the second kind only. To clarify this concept we consider the following equations: x u(x) = sin x + xtu(t)dt, (2.44)

(2.45)

(1 + x − t)u4 (t)dt,

0

(2.46)

and

0 x

0

(x − t)2 u(t)dt,

u(x) = x + u(x) =

1

x

u (x) =

xtu(t)dt,

u (0) = 0.

u(0) = 1,

(2.47)

0

The ﬁrst two equations are inhomogeneous because f (x) = sin x and f (x) = x, whereas the last two equations are homogeneous because f (x) = 0 for each equation. We usually use speciﬁc approaches for homogeneous equations, and other methods are used for inhomogeneous equations.

Exercises 2.3 Classify the following equations as Fredholm, or Volterra, linear or nonlinear, and homogeneous or inhomogeneous x 1 1. u(x) = 1 + (x − t)2 u(t)dt 2. u(x) = cosh x + (x − t)u(t)dt 3. u(x) =

0

x 0

(2 + x − t)u(t)dt

4. u(x) = λ

0

1

−1

2

t u(t)dt

42

2 Introductory Concepts of Integral Equations

5. u(x) = 1 + x + 7. u (x) = 1 +

1 0

x

0

(x − t)

1 dt 1 + u2

(x − t)u(t)dt, u(0) = 1

6. u(x) = 1 + 8. u (x) =

x

0

1

0

u3 (t)dt

(x − t)u(t)dt, u(0) = 0

2.4 Origins of Integral Equations Integral and integro-diﬀerential equations arise in many scientiﬁc and engineering applications. Volterra integral equations and Volterra integrodiﬀerential equations can be obtained from converting initial value problems with prescribed initial values. However, Fredholm integral equations and Fredholm integro-diﬀerential equations can be derived from boundary value problems with given boundary conditions. It is important to point out that converting initial value problems to Volterra integral equations, and converting Volterra integral equations to initial value problems are commonly used in the literature. This will be explained in detail in the coming section. However, converting boundary value problems to Fredholm integral equations, and converting Fredholm integral equations to equivalent boundary value problems are rarely used. The conversion techniques will be examined and illustrated examples will be presented. In what follows we will examine the steps that we will use to obtain these integral and integro-diﬀerential equations.

2.5 Converting IVP to Volterra Integral Equation In this section, we will study the technique that will convert an initial value problem (IVP) to an equivalent Volterra integral equation and Volterra integro-diﬀerential equation as well [3]. For simplicity reasons, we will apply this process to a second order initial value problem given by y (x) + p(x)y (x) + q(x)y(x) = g(x) subject to the initial conditions:

(2.48)

y(0) = α, y (0) = β, (2.49) where α and β are constants. The functions p(x) and q(x) are analytic functions, and g(x) is continuous through the interval of discussion. To achieve our goal we ﬁrst set (2.50) y (x) = u(x), where u(x) is a continuous function. Integrating both sides of (2.50) from 0 to x yields x y (x) − y (0) = u(t)dt, (2.51) 0

2.5 Converting IVP to Volterra Integral Equation

or equivalently y (x) = β +

43

x

u(t)dt.

(2.52)

0

Integrating both sides of (2.52) from 0 to x yields x x u(t)dtdt, y(x) − y(0) = βx + 0

or equivalently

(2.53)

0

x

y(x) = α + βx + 0

(x − t)u(t)dt,

(2.54)

obtained upon using the formula that reduce double integral to a single integral that was discussed in the previous chapter. Substituting (2.50), (2.52), and (2.54) into the initial value problem (2.48) yields the Volterra integral equation: x x u(t)dt + q(x) α + βx + (x − t)u(t)dt = g(x). u(x) + p(x) β + 0

0

(2.55) The last equation can be written in the standard Volterra integral equation form: x u(x) = f (x) − K(x, t)u(t)dt, (2.56) 0

where K(x, t) = p(x) + q(x)(x − t),

(2.57)

f (x) = g(x) − [βp(x) + αq(x) + βxq(x)] .

(2.58)

and It is interesting to point out that by diﬀerentiating Volterra equation (2.56) with respect to x, using Leibnitz rule, we obtain an equivalent Volterra integro-diﬀerential equation in the form: x ∂K(x, t) u (x) + K(x, x)u(x) = f (x) − u(t)dt, u(0) = f (0). (2.59) ∂x 0 The technique presented above to convert initial value problems to equivalent Volterra integral equations can be generalized by considering the general initial value problem: (2.60) y (n) + a1 (x)y (n−1) + · · · + an−1 (x)y + an (x)y = g(x), subject to the initial conditions y(0) = c0 , y (0) = c1 , y (0) = c2 , . . . , y (n−1) (0) = cn−1 . (2.61) We assume that the functions ai (x), 1 i n are analytic at the origin, and the function g(x) is continuous through the interval of discussion. Let u(x) be a continuous function on the interval of discussion, and we consider the transformation: y (n) (x) = u(x). (2.62) Integrating both sides with respect to x gives

44

2 Introductory Concepts of Integral Equations

y

(n−1)

(x) = cn−1 +

x

u(t)dt.

(2.63)

0

Integrating again both sides with respect to x yields x x (n−2) y (x) = cn−2 + cn−1 x + u(t)dtdt 0 x 0 (x − t)u(t)dt, = cn−2 + cn−1 x +

(2.64)

0

obtained by reducing the double integral to a single integral. Proceeding as before we ﬁnd x x x 1 y (n−3) (x) = cn−3 + cn−2 x + cn−1 x2 + u(t)dtdtdt 2 0 0 0 1 1 x (x − t)2 u(t)dt. (2.65) = cn−3 + cn−2 x + cn−1 x2 + 2 2 0 Continuing the integration process leads to x n−1 ck 1 xk + (x − t)n−1 u(t)dt. (2.66) y(x) = k! (n − 1)! 0 k=0

Substituting (2.62)–(2.66) into (2.60) gives x K(x, t)u(t)dt, u(x) = f (x) −

(2.67)

0

where K(x, t) =

n k=1

and f (x) = g(x) −

n j=1

an (x − t)k−1 , (k − 1)! aj

j cn−k j−k x (j − k)!

(2.68) .

(2.69)

k=1

Notice that the Volterra integro-diﬀerential equation can be obtained by differentiating (2.67) as many times as we like, and by obtaining the initial conditions of each resulting equation. The following examples will highlight the process to convert initial value problem to an equivalent Volterra integral equation. Example 2.1 Convert the following initial value problem to an equivalent Volterra integral equation: 2 y (x) − 2xy(x) = ex , y(0) = 1. (2.70) We ﬁrst set

y (x) = u(x).

(2.71)

Integrating both sides of (2.71), using the initial condition y(0) = 1 gives x u(t)dt, (2.72) y(x) − y(0) = 0

2.5 Converting IVP to Volterra Integral Equation

or equivalently

45

x

y(x) = 1 +

u(t)dt,

(2.73)

0

Substituting (2.71) and (2.73) into (2.70) gives the equivalent Volterra integral equation: x 2 u(t)dt. (2.74) u(x) = 2x + ex + 2x 0

Example 2.2 Convert the following initial value problem to an equivalent Volterra integral equation: y (x) − y(x) = sin x, y(0) = 0, y (0) = 0. (2.75) Proceeding as before, we set y (x) = u(x). (2.76) Integrating both sides of (2.76), using the initial condition y (0) = 0 gives x u(t)dt. (2.77) y (x) = 0

Integrating (2.77) again, using the initial condition y(0) = 0 yields x x x y(x) = u(t)dtdt = (x − t)u(t)dt, 0

0

(2.78)

0

obtained upon using the rule to convert double integral to a single integral. Inserting (2.76)–(2.78) into (2.70) leads to the following Volterra integral equation: x

u(x) = sin x + 0

(x − t)u(t)dt.

(2.79)

Example 2.3 Convert the following initial value problem to an equivalent Volterra integral equation: y − y − y + y = 0, We ﬁrst set

y(0) = 1,

y (0) = 2,

y (0) = 3.

y (x) = u(x),

(2.80) (2.81)

where by integrating both sides of (2.81) and using the initial condition y (0) = 3 we obtain x u(t)dt. (2.82) y = 3 + 0

Integrating again and using the initial condition y (0) = 2 we ﬁnd x x x y (x) = 2 + 3x + u(t)dtdt = 2 + 3x + (x − t)u(t)dt. 0

0

Integrating again and using y(0) = 1 we obtain

0

(2.83)

46

2 Introductory Concepts of Integral Equations

x x x 3 2 y(x) = 1 + 2x + x + u(t)dtdtdt 2 0 x0 0 3 1 = 1 + 2x + x2 + (x − t)2 u(t)dt. (2.84) 2 2 0 Notice that in (2.83) and (2.84) the multiple integrals were reduced to single integrals as used before. Substituting (2.81) – (2.84) into (2.80) leads to the Volterra integral equation: x 1 3 2 [1 + (x − t) − (x − t)2 ]u(t)dt. (2.85) u(x) = 4 + x + x + 2 2 0 Remark We can also show that if y (iv) (x) = u(x), then x u(t)dt y (x) = y (0) + 0 x y (x) = y (0) + xy (0) + (x − t)u(t)dt 0 1 2 1 x y (x) = y (0) + xy (0) + x y (0) + (x − t)2 u(t)dt 2 2 0 1 1 1 x (x − t)3 u(t)dt. y(x) = y(0) + xy (0) + x2 y (0) + x3 y (0) + 2 6 6 0 (2.86) This process can be generalized to any derivative of a higher order. In what follows we summarize the relation between derivatives of y(x) and u(x): Table 2.1 The relation between derivatives of y(x) and u(x) y (n) (x)

Integral Equations

y (x) = u(x)

y(x) = y(0) +

y (x) = u(x)

y (x) = y (0) +

y (x) = y (0) +

x

u(t)dt

0

x

0

y (x) = y (0) + xy (0) + y(x) = y(0) + xy (0) +

u(t)dt

0

y(x) = y(0) + xy (0) +

y (x) = u(x)

x

(x − t)u(t)dt

x

u(t)dt

0

x 0

(x − t)u(t)dt

1 2 1 x y (0) + 2 2

x

0

(x − t)2 u(t)dt

2.5 Converting IVP to Volterra Integral Equation

47

2.5.1 Converting Volterra Integral Equation to IVP A well-known method for solving Volterra integral and Volterra integrodiﬀerential equation, that we will use in the forthcoming chapters, converts these equations to equivalent initial value problems. The method is achieved simply by diﬀerentiating both sides of Volterra equations [6] with respect to x as many times as we need to get rid of the integral sign and come out with a diﬀerential equation. The conversion of Volterra equations requires the use of Leibnitz rule for diﬀerentiating the integral at the right hand side. The initial conditions can be obtained by substituting x = 0 into u(x) and its derivatives. The resulting initial value problems can be solved easily by using ODEs methods that were summarized in Chapter 1. The conversion process will be illustrated by discussing the following examples. Example 2.4 Find the initial value problem equivalent to the Volterra integral equation: x x u(t)dt. (2.87) u(x) = e + 0

Diﬀerentiating both sides of (2.87) and using Leibnitz rule we ﬁnd u (x) = ex + u(x).

(2.88)

It is clear that there is no need for diﬀerentiating again because we got rid of the integral sign. To determine the initial condition, we substitute x = 0 into both sides of (2.87) to ﬁnd u(0) = 1. This in turn gives the initial value problem: u (x) − u(x) = ex , u(0) = 1. (2.89) Notice that the resulting ODE is a linear inhomogeneous equation of ﬁrst order. Example 2.5 Find the initial value problem equivalent to the Volterra integral equation: x (x − t)u(t)dt. (2.90) u(x) = x2 + 0

Diﬀerentiating both sides of (2.90) and using Leibnitz rule we ﬁnd x u (x) = 2x + u(t)dt.

(2.91)

0

To get rid of the integral sign we should diﬀerentiate (2.91) and by using Leibnitz rule we obtain the second order ODE: (2.92) u (x) = 2 + u(x). To determine the initial conditions, we substitute x = 0 into both sides of (2.90) and (2.91) to ﬁnd u(0) = 0 and u (0) = 0 respectively. This in turn gives the initial value problem: (2.93) u (x) − u(x) = 2, u(0) = 0, u (0) = 0.

48

2 Introductory Concepts of Integral Equations

Notice that the resulting ODE is a second order inhomogeneous equation. Example 2.6 Find the initial value problem equivalent to the Volterra integral equation: 1 x u(x) = sin x − (x − t)2 u(t)dt. (2.94) 2 0 Diﬀerentiating both sides of the integral equation three times to get rid of the integral sign to ﬁnd x (x − t)u(t)dt, u (x) = cos x − 0 x (2.95) u (x) = − sin x − u(t)dt, 0

u (x) = − cos x − u(x). Substituting x = 0 into (2.94) and into the ﬁrst two integro-diﬀerential equations in (2.95) gives the initial conditions: u(0) = 0, u (0) = 1, u (0) = 0. (2.96) In view of the last results, the initial value problem equivalent to the Volterra integral equation (2.94) is a third order inhomogeneous ODE given by u (x) + u(x) = − cos x,

u(0) = 0,

u (0) = 1,

u (0) = 0.

(2.97)

Exercises 2.5 Convert each of the following IVPs in 1–8 to an equivalent Volterra integral equation: 2

1. y − 4y = 0, y(0) = 1

2. y + 4xy = e−2x , y(0) = 0

3. y + 4y = 0, y(0) = 0, y (0) = 1

4. y − 6y + 8y = 1, y(0) = 1, y (0) = 1

5. y − y = 0, y(0) = 2, y (0) = y (0) = 1 6. y − 2y + y = x, y(0) = 1, y (0) = 0, y (0) = 1 7. y (iv) − y = 1, y(0) = y (0) = 0, y (0) = y (0) = 1 8. y (iv) + y + y = x, y(0) = y (0) = 1, y (0) = y (0) = 0 Convert each of the following Volterra integral equation in 9–16 to an equivalent IVP: x x 9. u(x) = x + 2 u(t)dt 10. u(x) = 1 + ex − u(t)dt 0

11. u(x) = 1 + x2 +

x

0

13. u(x) = 1 − cos x + 2 15. u(x) = 1 + 2

x 0

(x − t)u(t)dt

x

0

12. u(x) = sin x −

0 x

(x − t)u(t)dt

0

(x − t)2 u(t)dt 14. u(x) = 2 + sinh x +

(x − t)3 u(t)dt

16. u(x) = 1 + ex +

x

0

x

0

(x − t)2 u(t)dt

(1 + x − t)3 u(t)dt

2.6 Converting BVP to Fredholm Integral Equation

49

2.6 Converting BVP to Fredholm Integral Equation In this section, we will present a method that will convert a boundary value problem to an equivalent Fredholm integral equation. The method is similar to the method that was presented in the previous section for converting Volterra equation to IVP, with the exception that boundary conditions will be used instead of initial values. In this case we will determine another initial condition that is not given in the problem. The technique requires more work if compared with the initial value problems when converted to Volterra integral equations. For this reason, the technique that will be presented is rarely used. Without loss of generality, we will present two speciﬁc distinct boundary value problems (BVPs) to derive two distinct formulas that can be used for converting BVP to an equivalent Fredholm integral equation. Type I We ﬁrst consider the following boundary value problem: y (x) + g(x)y(x) = h(x), 0 < x < 1,

(2.98)

with the boundary conditions: y(0) = α,

(2.99)

y(1) = β.

We start as in the previous section and set y (x) = u(x).

(2.100)

Integrating both sides of (2.100) from 0 to x we obtain x x y (t)dt = u(t)dt,

(2.101)

0

0

that gives y (x) = y (0) +

x

u(t)dt,

(2.102)

0

where the initial condition y (0) is not given in a boundary value problem. The condition y (0) will be determined later by using the boundary condition at x = 1. Integrating both sides of (2.102) from 0 to x gives x x u(t)dt, (2.103) y(x) = y(0) + xy (0) + 0

or equivalently y(x) = α + xy (0) +

0

0

x

(x − t)u(t)dt,

(2.104)

obtained upon using the condition y(0) = α and by reducing double integral to a single integral. To determine y (0), we substitute x = 1 into both sides of (2.104) and using the boundary condition at y(1) = β we ﬁnd 1 (1 − t)u(t)dt, (2.105) y(1) = α + y (0) + 0

that gives

50

2 Introductory Concepts of Integral Equations

β = α + y (0) +

1

0

This in turn gives y (0) = (β − α) −

0

(1 − t)u(t)dt.

1

(1 − t)u(t)dt.

(2.106)

(2.107)

Substituting (2.107) into (2.104) gives x 1 x(1 − t)u(t)dt + (x − t)u(t)dt. y(x) = α + (β − α)x −

(2.108)

Substituting (2.100) and (2.108) into (2.98) yields 1 xg(x)(1 − t)u(t)dt u(x) + αg(x) + (β − α)xg(x) − 0 x g(x)(x − t)u(t)dt = h(x). +

(2.109)

From calculus we can use the formula: x 1 1 (·) = (·) + (·),

(2.110)

0

0

0

0

0

x

to carry Eq. (2.109) to

x

u(x) = h(x) − αg(x) − (β − α)xg(x) − g(x) (x − t)u(t)dt 0 x 1 (1 − t)u(t)dt + (1 − t)u(t)dt , (2.111) +xg(x) 0

that gives

u(x) = f (x) + 0

x

x

t(1 − x)g(x)u(t)dt +

1

x(1 − t)g(x)u(t)dt,

(2.112)

x

that leads to the Fredholm integral equation: 1 K(x, t)u(t)dt, u(x) = f (x) +

(2.113)

0

where f (x) = h(x) − αg(x) − (β − α)xg(x),

(2.114)

and the kernel K(x, t) is given by t(1 − x)g(x), for 0 t x, (2.115) K(x, t) = x(1 − t)g(x), for x t 1. An important conclusion can be made here. For the speciﬁc case where y(0) = y(1) = 0 which means that α = β = 0, i.e. the two boundaries of a moving string are ﬁxed, it is clear that f (x) = h(x) in this case. This means that the resulting Fredholm equation in (2.113) is homogeneous or inhomogeneous if the boundary value problem in (2.98) is homogeneous or

2.6 Converting BVP to Fredholm Integral Equation

51

inhomogeneous respectively when α = β = 0. The techniques presented above will be illustrated by the following two examples. Example 2.7 Convert the following BVP to an equivalent Fredholm integral equation: y (x) + 9y(x) = cos x, y(0) = y(1) = 0. (2.116) We can easily observe that α = β = 0, g(x) = 9 and h(x) = cos x. This in turn gives f (x) = cos x. (2.117) Substituting this into (2.113) gives the Fredholm integral equation: 1 K(x, t)u(t)dt, (2.118) u(x) = cos x + 0

where the kernel K(x, t) is given by 9t(1 − x), K(x, t) = 9x(1 − t),

for 0 t x, for x t 1.

(2.119)

Example 2.8 Convert the following BVP to an equivalent Fredholm integral equation: (2.120) y (x) + xy(x) = 0, y(0) = 0, y(1) = 2. Recall that this is a boundary value problem because the conditions are given at the boundaries x = 0 and x = 1. Moreover, the coeﬃcient of y(x) is a variable and not a constant. We can easily observe that α = 0, β = 2, g(x) = x and h(x) = 0. This in turn gives f (x) = −2x2 . (2.121) Substituting this into (2.113) gives the Fredholm integral equation: 1 K(x, t)u(t)dt, (2.122) u(x) = −2x2 + 0

where the kernel K(x, t) is given by tx(1 − x), K(x, t) = x2 (1 − t),

for 0 t x, for x t 1.

(2.123)

Type II We next consider the following boundary value problem: y (x) + g(x)y(x) = h(x), with the boundary conditions: y(0) = α1 , We again set

0 < x < 1,

y (1) = β1 .

y (x) = u(x).

(2.124) (2.125) (2.126)

52

2 Introductory Concepts of Integral Equations

Integrating both sides of (2.126) from 0 to x we obtain x x y (t)dt = u(t)dt, 0

(2.127)

0

that gives y (x) = y (0) +

x

u(t)dt,

(2.128)

0

where the initial condition y (0) is not given. The condition y (0) will be derived later by using the boundary condition at y (1) = β1 . Integrating both sides of (2.128) from 0 to x gives x x u(t)dtdt, (2.129) y(x) = y(0) + xy (0) + 0

or equivalently y(x) = α1 + xy (0) +

0

x 0

(x − t)u(t)dt,

(2.130)

obtained upon using the condition y(0) = α1 and by reducing double integral to a single integral. To determine y (0), we ﬁrst diﬀerentiate (2.130) with respect to x to get x

y (x) = y (0) +

u(t)dt,

(2.131)

0

where by substituting x = 1 into both sides of (2.131) and using the boundary condition at y (1) = β1 we ﬁnd 1 y (1) = y (0) + u(t)dt, (2.132) 0

that gives

y (0) = β1 −

1

u(t)dt.

(2.133)

0

Using (2.133) into (2.130) gives x 1 u(t)dt + (x − t)u(t)dt. y(x) = α1 + x β1 − 0

(2.134)

0

Substituting (2.126) and (2.134) into (2.124) yields 1 x u(x) + α1 g(x) + β1 xg(x) − xg(x)u(t)dt + g(x)(x − t)u(t)dt = h(x). 0

0

(2.135)

From calculus we can use the formula: x 1 1 (·) = (·) + (·), 0

0

(2.136)

x

to carry Eq. (2.135) to u(x) = h(x) − (α1 + β1 x)g(x) x +xg(x) u(t)dt + 0

1

x

x u(t)dt − g(x) (x − t)u(t)dt. (2.137) 0

2.6 Converting BVP to Fredholm Integral Equation

The last equation can be written as x u(x) = f (x) + tg(x)u(t)dt + 0

53

1

xg(x)u(t)dt,

(2.138)

x

that leads to the Fredholm integral equation: 1 K(x, t)u(t)dt, u(x) = f (x) +

(2.139)

0

where f (x) = h(x) − (α1 + β1 x)g(x), and the kernel K(x, t) is given by tg(x), K(x, t) = xg(x),

for 0 t x, for x t 1.

(2.140)

(2.141)

An important conclusion can be made here. For the speciﬁc case where y(0) = y (1) = 0 which means that α1 = β1 = 0, it is clear that f (x) = h(x) in this case. This means that the resulting Fredholm equation in (2.139) is homogeneous or inhomogeneous if the boundary value problem in (2.124) is homogeneous or inhomogeneous respectively. The second type of conversion that was presented above will be illustrated by the following two examples. Example 2.9 Convert the following BVP to an equivalent Fredholm integral equation: (2.142) y (x) + y(x) = 0, y(0) = y (1) = 0. We can easily observe that α1 = β1 = 0, g(x) = 1 and h(x) = 0. This in turn gives f (x) = 0. (2.143) Substituting this into (2.139) gives the homogeneous Fredholm integral equation: 1

u(x) =

K(x, t)u(t)dt,

(2.144)

0

where the kernel K(x, t) is given by t, K(x, t) = x,

for 0 t x, for x t 1.

(2.145)

Example 2.10 Convert the following BVP to an equivalent Fredholm integral equation: (2.146) y (x) + 2y(x) = 4, y(0) = 0, y (1) = 1. We can easily observe that α1 = 0, β1 = 1, g(x) = 2 and h(x) = 4. This in turn gives f (x) = 4 − 2x. (2.147)

54

2 Introductory Concepts of Integral Equations

Substituting this into (2.139) gives the inhomogeneous Fredholm integral equation: 1 u(x) = 4 − 2x + K(x, t)u(t)dt, (2.148) 0

where the kernel K(x, t) is given by 2t, K(x, t) = 2x,

for 0 t x, for x t 1.

(2.149)

2.6.1 Converting Fredholm Integral Equation to BVP In a previous section, we presented a technique to convert Volterra integral equation to an equivalent initial value problem. In a similar manner, we will present another technique that will convert Fredholm integral equation to an equivalent boundary value problem (BVP). In what follows we will examine two types of problems: Type I We ﬁrst consider the Fredholm integral equation given by 1 K(x, t)u(t)dt, u(x) = f (x) +

(2.150)

0

where f (x) is a given function, and the kernel K(x, t) is given by t(1 − x)g(x), for 0 t x, K(x, t) = x(1 − t)g(x), for x t 1.

(2.151)

For simplicity reasons, we may consider g(x) = λ where λ is constant. Equation (2.150) can be written as x 1 u(x) = f (x) + λ t(1 − x)u(t)dt + λ x(1 − t)u(t)dt, (2.152) 0

or equivalently u(x) = f (x) + λ(1 − x)

x

x

1

tu(t)dt + λx 0

(1 − t)u(t)dt.

(2.153)

x

Each term of the last two terms at the right side of (2.153) is a product of two functions of x. Diﬀerentiating both sides of (2.153), using the product rule of diﬀerentiation and using Leibnitz rule we obtain x tu(t) u (x) = f (x) + λx(1 − x)u(x) − λ 0 1 −λx(1 − x)u(x) + λ (1 − t)u(t)dt (2.154) x x 1 = f (x) − λ tu(t) + λ (1 − t)u(t)dt. 0

x

2.6 Converting BVP to Fredholm Integral Equation

55

To get rid of integral signs, we diﬀerentiate both sides of (2.154) again with respect to x to ﬁnd that u (x) = f (x) − λxu(x) − λ(1 − x)u(x), (2.155) that gives the ordinary diﬀerential equations: (2.156) u (x) + λu(x) = f (x). The related boundary conditions can be obtained by substituting x = 0 and x = 1 in (2.153) to ﬁnd that u(0) = f (0), u(1) = f (1). (2.157) Combining (2.156) and (2.157) gives the boundary value problem equivalent to the Fredholm equation (2.150). Recall that y (x) = u(x). Moreover, if g(x) is not a constant, we can proceed in a manner similar to the discussion presented above to obtain the boundary value problem. The technique above for type I will be explained by studying the following examples. Example 2.11 Convert the Fredholm integral equation 1 K(x, t)u(t)dt, u(x) = ex +

(2.158)

0

where the kernel K(x, t) is given by 9t(1 − x), for 0 t x, K(x, t) = 9x(1 − t), for x t 1, to an equivalent boundary value problem. The Fredholm integral equation can be written as x 1 x u(x) = e + 9(1 − x) tu(t)dt + 9x (1 − t)u(t)dt. 0

(2.160)

x

Diﬀerentiating (2.160) twice with respect to x gives x 1 tu(t)dt + 9 (1 − t)u(t)dt, u (x) = ex − 9 0

and

(2.159)

(2.161)

x

u (x) = ex − 9u(x).

(2.162)

This in turn gives the ODE: u (x) + 9u(x) = ex . The related boundary conditions are given by u(0) = f (0) = 1,

u(1) = f (1) = e,

obtained upon substituting x = 0 and x = 1 into (2.160). Example 2.12 Convert the Fredholm integral equation

(2.163) (2.164)

56

2 Introductory Concepts of Integral Equations

u(x) = x3 +

1

K(x, t)u(t)dt,

(2.165)

0

where the kernel K(x, t) is given by 4t(1 − x), for 0 t x, K(x, t) = 4x(1 − t), for x t 1, to an equivalent boundary value problem. The Fredholm integral equation can be written as x 1 3 u(x) = x + 4(1 − x) tu(t)dt + 4x (1 − t)u(t)dt. 0

(2.166)

(2.167)

x

Proceeding as before we ﬁnd u (x) = 6x − 4u(x).

(2.168)

This in turn gives the ODE: u (x) + 4u(x) = 6x,

(2.169)

with the related boundary conditions: u(0) = f (0) = 0, u(1) = f (1) = 1.

(2.170)

Type II We next consider the Fredholm integral equation given by 1 K(x, t)u(t)dt, u(x) = f (x) +

(2.171)

0

where f (x) is a given function, and the kernel K(x, t) is given by tg(x), for 0 t x, K(x, t) = xg(x), for x t 1.

(2.172)

For simplicity reasons, we again consider g(x) = λ where λ is constant. Equation (2.171) can be written as x 1 u(x) = f (x) + λ tu(t)dt + λx u(t)dt. (2.173) 0

x

Each integral at the right side of (2.173) is a product of two functions of x. Diﬀerentiating both sides of (2.173), using the product rule of diﬀerentiation and using Leibnitz rule we obtain 1 u (x) = f (x) + λ u(t)dt. (2.174) x

To get rid of integral signs, we diﬀerentiate again with respect to x to ﬁnd that (2.175) u (x) = f (x) − λu(x), that gives the ordinary diﬀerential equations. Also change equations to equation (2.176) u (x) + λu(x) = f (x).

2.6 Converting BVP to Fredholm Integral Equation

57

Notice that the boundary condition u(1) in this case cannot be obtained from (2.173). Therefore, the related boundary conditions can be obtained by substituting x = 0 and x = 1 in (2.173) and (2.174) respectively to ﬁnd that u(0) = f (0), u (1) = f (1). (2.177) Combining (2.176) and (2.177) gives the boundary value problem equivalent to the Fredholm equation (2.171). Recall that y (x) = u(x). Moreover, if g(x) is not a constant, we can proceed in a manner similar to the discussion presented above to obtain the boundary value problem. The approach presented above for type II will be illustrated by studying the following examples. Example 2.13 Convert the Fredholm integral equation: 1 K(x, t)u(t)dt, u(x) = ex +

(2.178)

0

where the kernel K(x, t) is given by 4t, for 0 t x, K(x, t) = 4x, for x t 1, to an equivalent boundary value problem. The Fredholm integral equation can be written as x 1 x u(x) = e + 4 tu(t)dt + 4x u(t)dt. 0

(2.179)

(2.180)

x

Diﬀerentiating (2.180) twice with respect to x gives 1 u(t)dt, u (x) = ex + 4

(2.181)

x

and

u (x) = ex − 4u(x).

(2.182)

This in turn gives the ODE: u (x) + 4u(x) = ex . The related boundary conditions are given by

(2.183)

u(0) = f (0) = 1, u (1) = f (1) = e, (2.184) obtained upon substituting x = 0 and x = 1 into (2.180) and (2.181) respectively. Recall that the boundary condition u(1) cannot obtained in this case. Example 2.14 Convert the Fredholm integral equation 1 K(x, t)u(t)dt, u(x) = x2 + 0

where the kernel K(x, t) is given by

(2.185)

58

2 Introductory Concepts of Integral Equations

K(x, t) =

for 0 t x,

6t,

6x, for x t 1 to an equivalent boundary value problem. The Fredholm integral equation can be written as x 1 u(x) = x2 + 6 tu(t)dt + 6x u(t)dt. 0

(2.186)

(2.187)

x

Proceeding as before we ﬁnd u (x) = 2x + 6

1

u(t)dt.

(2.188)

x

and

u (x) + 6u(x) = 2,

(2.189)

with the related boundary conditions u(0) = f (0) = 0,

u (1) = f (1) = 2.

(2.190)

Exercises 2.6 Convert each of the following BVPs in 1–8 to an equivalent Fredholm integral equation: 1. y + 4y = 0, 0 < x < 1, y(0) = y(1) = 0 2. y + xy = 0, y(0) = y(1) = 0 3. y + 2y = x, 0 < x < 1, y(0) = 1, y(1) = 0 4. y + 3xy = 4, 0 < x < 1, y(0) = 0, y(1) = 0 5. y + 4y = 0, 0 < x < 1, y(0) = 0, y (1) = 0 6. y + xy = 0, y(0) = 0, y (1) = 0 7. y + 4y = x, 0 < x < 1, y(0) = 1, y (1) = 0 8. y + 4xy = 2, 0 < x < 1, y(0) = 0, y (1) = 1 Convert each of the following Fredholm integral equation in 9–16 to an equivalent BVP: 1 3t(1 − x), for 0 t x K(x, t)u(t)dt, K(x, t) = 9. u(x) = e2x + 3x(1 − t), for x t 1 0 1 t(1 − x), for 0 t x K(x, t)u(t)dt, K(x, t) = 10. u(x) = 3x2 + x(1 − t), for x t 1 0 1 6t(1 − x), for 0 t x 11. u(x) = cos x + K(x, t)u(t)dt, K(x, t) = 6x(1 − t), for x t 1 0 1 4t(1 − x), for 0 t x K(x, t)u(t)dt, K(x, t) = 12. u(x) = sinh x + 4x(1 − t), for x t 1 0

2.7 Solution of an Integral Equation 13. u(x) = e3x + 14. u(x) = x4 +

59

1

0

K(x, t)u(t)dt, K(x, t) =

1

t,

for 0 t x

x,

for x t 1

6t,

for 0 t x

K(x, t)u(t)dt, K(x, t) =

6x, for x t 1 1 4t, for 0 t x 2 K(x, t)u(t)dt, K(x, t) = 15. u(x) = 2x + 3 + 4x, for x t 1 0 1 2t, for 0 t x 16. u(x) = ex + 1 + K(x, t)u(t)dt, K(x, t) = 2x, for xt1 0 0

2.7 Solution of an Integral Equation A solution of a diﬀerential or an integral equation arises in any of the following two types: 1). Exact solution: The solution is called exact if it can be expressed in a closed form, such as a polynomial, exponential function, trigonometric function or the combination of two or more of these elementary functions. Examples of exact solutions are as follows: u(x) = x + ex , u(x) = sin x + e2x ,

(2.191)

u(x) = 1 + cosh x + tan x, and many others. 2). Series solution: For concrete problems, sometimes we cannot obtain exact solutions. In this case we determine the solution in a series form that may converge to exact soliton if such a solution exists. Other series may not give exact solution, and in this case the obtained series can be used for numerical purposes. The more terms that we determine the higher accuracy level that we can achieve. A solution of an integral equation or integro-diﬀerential equation is a function u(x) that satisﬁes the given equation. In other words, the obtained solution u(x) must satisfy both sides of the examined equation. The following examples will be examined to explain the meaning of a solution. Example 2.15 Show that u(x) = sinh x is a solution of the Volterra integral equation: x (x − t)u(t)dt. (2.192) u(x) = x + 0

Substituting u(x) = sinh x in the right hand side (RHS) of (2.192) yields

60

2 Introductory Concepts of Integral Equations

RHS = x + 0

x

(x − t) sinh tdt

= x + (sinh t − t)|x0 = sinh x = u(x) = LHS.

(2.193)

Example 2.16 Show that u(x) = sec2 x is a solution of the Fredholm integral equation π 1 1 4 2 u(x) = − + sec x + u(t)dt. (2.194) 2 2 0 Substituting u(x) = sec2 x in the right hand side of (2.194) gives π 1 4 1 2 u(t)dt RHS = − + sec x + 2 2 0 π 1 1 = − + sec2 x + (tan t) 04 2 2 = sec2 x = u(x) = LHS. (2.195) Example 2.17 Show that u(x) = sin x is a solution of the Volterra integro-diﬀerential equation: x u (x) = 1 − u(t)dt. (2.196) 0

Proceeding as before, and using u(x) = sin x into both sides of (2.196) we ﬁnd LHS = u (x) = cos x, x x sin tdt = 1 − (− cos t) 0 = cos x. (2.197) RHS = 1 − 0

Example 2.18 Show that u(x) = x + ex is a solution of the Fredholm integro-diﬀerential equation: 1 4 x u (x) = e − x + xtu(t)dt. (2.198) 3 0 Substituting u(x) = x + ex into both sides of (2.198) we ﬁnd LHS = u (x) = ex , 1 4 x RHS = e − x + x t(t + et )dt 3 0 1 1 3 4 = ex − x + x t + tet − et = ex . 3 3 0

(2.199)

2.7 Solution of an Integral Equation

61

Example 2.19 Show that u(x) = cos x is a solution of the Volterra-Fredholm integral equation: x π2 u(x) = cos x − x + u(t)dtdt. (2.200) 0

0

Proceeding as before, and using u(x) = cos x into both sides of (2.200) we ﬁnd LHS = cos x, RHS = cos x − x +

0

x

π 2

cos tdt = cos x.

(2.201)

0

Example 2.20 Show that u(x) = ex is a solution of the Fredholm integral equation of the ﬁrst kind: 1 2 2 ex +1 − 1 = ex t u(t)dt. (2.202) x2 + 1 0 Proceeding as before, and using u(x) = ex into the right side of (2.202) we ﬁnd t=1 1 2 2 e(x +1)t ex +1 − 1 (x2 +1)t = LHS. (2.203) e dt = 2 = RHS = x +1 x2 + 1 0 t=0

Example 2.21 Show that u(x) = x is a solution of the nonlinear Fredholm integral equation π 1 1 1 u(x) = x − + dt. (2.204) 12 3 0 1 + u2 (t) Using u(x) = x into the right side of (2.204) we ﬁnd t=1 1 1 1 1 π π + + tan−1 t t=0 dt = x − RHS = x − 2 12 3 0 1 + t 12 3 π 1 π = x− + − 0 = x = LHS. (2.205) 12 3 4 Example 2.22 Find f (x) if u(x) = x2 + x3 is a solution of the Fredholm integral equation 5 1 u(x) = f (x) + (xt2 + x2 t)u(t)dt. (2.206) 2 −1 Using u(x) = x2 + x3 into both sides of (2.206) we ﬁnd LHS = x2 + x3 , 5 1 (xt2 + x2 t)u(t)dt = f (x) + x2 + x. RHS = f (x) + 2 −1 Equating the left and right sides gives

(2.207)

62

2 Introductory Concepts of Integral Equations

f (x) = x3 − x.

(2.208)

Exercises 2.7 In Exercises 1–4, show that the given function u(x) is a solution of the corresponding Fredholm integral equation: π 1 2 sin xu(t)dt, u(x) = sin x + cos x 1. u(x) = cos x + 2 0 1 1 5 1 2. u(x) = e2x+ 3 − e2x− 3 t u(t)dt, u(x) = e2x 3 0 1 3. u(x) = x + (x4 − t4 )u(t)dt, − 1 x 1, u(x) = x −1

4. u(x) = x + (1 − x)ex +

1 0

x2 et(x−1) u(t)dt, u(x) = ex

In Exercises 5–8, show that the given function u(x) is a solution of the corresponding Volterra integral equation: 1 x 5. u(x) = 1 + u(t)dt, u(x) = e2x 2 0 x 6. u(x) = 4x + sin x + 2x2 − cos x + 1 − u(t)dt, u(x) = 4x + sin x 1 7. u(x) = 1 − x2 − 2

0

x 0

(x − t)u(t)dt, u(x) = 2 cos x − 1

8. u(x) = 1 + 2x + sin x + x2 − cos x −

x 0

u(t)dt, u(x) = 2x + sin x

In Exercises 9–12, show that the given function u(x) is a solution of the corresponding Fredholm integro-diﬀerential equation: 1 1 xu(t)dt, u(0) = 0, u(x) = xex 9. u (x) = xex + ex − x + 2 0 1 10. u (x) = ex + (e − 1) − u(t)dt, u(0) = 1, u(x) = ex 11. u (x) = 1 − sin x −

0

12. u (x) = 1 + sin x −

π 2

0

π 2

0

tu(t)dt, u(0) = 0, u (0) = 1, u(x) = sin x (x − t)u(t)dt,

u(0) = 1, u (0) = 0, u (0) = −1, u(x) = cos x In Exercises 13–16, show that the given function u(x) is a solution of the corresponding Volterra integro-diﬀerential equation: x 13. u (x) = 2 + x + x2 − u(t)dt, u(0) = 1, u(x) = 1 + 2x 0

14. u (x) = x cos x − 2 sin x + 15. u (x) = 1 +

x 0

x 0

tu(t)dt, u(0) = 0, u (0) = 1, u(x) = sin x

(x − t)u(t)dt, u(0) = 1, u (0) = 0, u(x) = cosh x

References

63

16. u (x) = 1 − xe−x −

x

0

tu(t)dt, u(0) = 1, u (0) = −1, u(x) = e−x

In Exercises 17–24, ﬁnd the unknown if the solution of each equation is given: x 17. If u(x) = e4x is a solution of u(x) = f (x) + 16 (x − t)u(t)dt, ﬁnd f (x) 0

18. If u(x) = e2x is a solution of u(x) = e2x − α(e2 + 1)x + 19. If u(x) = sin x is a solution of u(x) = f (x) + sin x − 2

20. If u(x) = e−x is a solution of u(x) = 1 − α 21. If u(x) = ex is a solution of u(x) = f (x) +

0

24. If u(x) = 6x is a solution of u(x) = f (x) + x

1

0

x 0

xtu(t)dt, ﬁnd α

xu(t)dt, ﬁnd f (x)

(2u2 (t) + u(t))dt, ﬁnd f (x)

x

0

0

π 2

u2 (t)dtdt, ﬁnd f (x)

23. If u(x) = 2 + 12x2 is a solution of u (x) = f (x) + 20x − ﬁnd f (x)

0

tu(t)dt, ﬁnd α

4 π

22. If u(x) = sin x is a solution of u(x) = f (x) +

0

1

x

0 x

π 2

x

0

1

0

(x − t)u(t)dtdt,

(1 − t)u(t)dt−

(x − t)u(t)dtdt, ﬁnd f (x)

References 1. C.D. Green, Integral Equations Methods, Barnes and Noble, New York, (1969). 2. H. Hochstadt, Integral Equations,Wiley, New York, (1973). 3. A. Jerri, Introduction to Integral Equations with Applications,Wiley, New York, (1999). 4. R. Kanwal, Linear Integral Equations, Birkhauser, Boston, (1997). 5. R. Kress, Linear Integral Equations, Springer, Berlin, (1999). 6. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 7. K. Maleknejad and Y. Mahmoudi, Taylor polynomial solution of highorder nonlinear Volterra-Fredholm integro-diﬀerential equations, Appl. Math. Comput. 145 (2003) 641–653. 8. A.M. Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. Comput., 127 (2002) 405–414.

Chapter 3

Volterra Integral Equations

3.1 Introduction It was stated in Chapter 2 that Volterra integral equations arise in many scientiﬁc applications such as the population dynamics, spread of epidemics, and semi-conductor devices. It was also shown that Volterra integral equations can be derived from initial value problems. Volterra started working on integral equations in 1884, but his serious study began in 1896. The name integral equation was given by du Bois-Reymond in 1888. However, the name Volterra integral equation was ﬁrst coined by Lalesco in 1908. Abel considered the problem of determining the equation of a curve in a vertical plane. In this problem, the time taken by a mass point to slide under the inﬂuence of gravity along this curve, from a given positive height, to the horizontal axis is equal to a prescribed function of the height. Abel derived the singular Abel’s integral equation, a speciﬁc kind of Volterra integral equation, that will be studied in a forthcoming chapter. Volterra integral equations, of the ﬁrst kind or the second kind, are characterized by a variable upper limit of integration [1]. For the ﬁrst kind Volterra integral equations, the unknown function u(x) occurs only under the integral sign in the form: x

f (x) =

K(x, t)u(t)dt.

(3.1)

0

However, Volterra integral equations of the second kind, the unknown function u(x) occurs inside and outside the integral sign. The second kind is represented in the form: x u(x) = f (x) + λ K(x, t)u(t)dt. (3.2) 0

The kernel K(x, t) and the function f (x) are given real-valued functions, and λ is a parameter [2–4]. A variety of analytic and numerical methods, such as successive approximations method, Laplace transform method, spline collocation method, A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

66

3 Volterra Integral Equations

Runge-Kutta method, and others have been used to handle Volterra integral equations. In this text we will apply the recently developed methods, namely, the Adomian decomposition method (ADM), the modiﬁed decomposition method (mADM), and the variational iteration method (VIM) to handle Volterra integral equations. Some of the traditional methods, namely, successive approximations method, series solution method, and the Laplace transform method will be employed as well. The emphasis in this text will be on the use of these methods and approaches rather than proving theoretical concepts of convergence and existence. The theorems of uniqueness, existence, and convergence are important and can be found in the literature. The concern will be on the determination of the solution u(x) of the Volterra integral equation of ﬁrst and second kind.

3.2 Volterra Integral Equations of the Second Kind We will ﬁrst study Volterra integral equations of the second kind given by x u(x) = f (x) + λ K(x, t)u(t)dt. (3.3) 0

The unknown function u(x), that will be determined, occurs inside and outside the integral sign. The kernel K(x, t) and the function f (x) are given real-valued functions, and λ is a parameter. In what follows we will present the methods, new and traditional, that will be used.

3.2.1 The Adomian Decomposition Method The Adomian decomposition method (ADM) was introduced and developed by George Adomian in [5–7] and is well addressed in many references. A considerable amount of research work has been invested recently in applying this method to a wide class of linear and nonlinear ordinary diﬀerential equations, partial diﬀerential equations and integral equations as well. The Adomian decomposition method consists of decomposing the unknown function u(x) of any equation into a sum of an inﬁnite number of components deﬁned by the decomposition series ∞ u(x) = un (x), (3.4) n=0

or equivalently u(x) = u0 (x) + u1 (x) + u2 (x) + · · · ,

(3.5)

where the components un (x), n 0 are to be determined in a recursive manner. The decomposition method concerns itself with ﬁnding the components

3.2 Volterra Integral Equations of the Second Kind

67

u0 , u1 , u2 , . . . individually. As will be seen through the text, the determination of these components can be achieved in an easy way through a recurrence relation that usually involves simple integrals that can be easily evaluated. To establish the recurrence relation, we substitute (3.4) into the Volterra integral equation (3.3) to obtain ∞ x ∞ un (x) = f (x) + λ K(x, t) un (t) dt, (3.6) 0

n=0

n=0

or equivalently

u0 (x) + u1 (x) + u2 (x) + · · · = f (x) + λ

x 0

K(x, t) [u0 (t) + u1 (t) + · · · ] dt.

(3.7) The zeroth component u0 (x) is identiﬁed by all terms that are not included under the integral sign. Consequently, the components uj (x), j 1 of the unknown function u(x) are completely determined by setting the recurrence relation: u0 (x) = f (x), x (3.8) K(x, t)un (t)dt, n 0, un+1 (x) = λ 0

that is equivalent to u0 (x) = f (x), u2 (x) = λ

0

u1 (x) = λ

x

K(x, t)u1 (t)dt,

u3 (x) = λ

x

K(x, t)u0 (t)dt,

0

(3.9)

x 0

K(x, t)u2 (t)dt,

and so on for other components. In view of (3.9), the components u0 (x), u1 (x), u2 (x), u3 (x), . . . are completely determined. As a result, the solution u(x) of the Volterra integral equation (3.3) in a series form is readily obtained by using the series assumption in (3.4). It is clearly seen that the decomposition method converted the integral equation into an elegant determination of computable components. It was formally shown by many researchers that if an exact solution exists for the problem, then the obtained series converges very rapidly to that solution. The convergence concept of the decomposition series was thoroughly investigated by many researchers to conﬁrm the rapid convergence of the resulting series. However, for concrete problems, where a closed form solution is not obtainable, a truncated number of terms is usually used for numerical purposes. The more components we use the higher accuracy we obtain. Example 3.1 Solve the following Volterra integral equation: x u(t)dt. u(x) = 1 − 0

(3.10)

68

3 Volterra Integral Equations

We notice that f (x) = 1, λ = −1, K(x, t) = 1. Recall that the solution u(x) is assumed to have a series form given in (3.4). Substituting the decomposition series (3.4) into both sides of (3.10) gives x ∞ ∞ un (x) = 1 − un (t)dt, (3.11) 0

n=0

or equivalently

u0 (x) + u1 (x) + u2 (x) + · · · = 1 −

n=0

x 0

[u0 (t) + u1 (t) + u2 (t) + · · · ] dt. (3.12)

We identify the zeroth component by all terms that are not included under the integral sign. Therefore, we obtain the following recurrence relation: u0 (x) = 1, x (3.13) uk (t)dt, k 0, uk+1 (x) = − 0

so that u0 (x) = 1, u1 (x) = − u2 (x) = − u3 (x) = − u4 (x) = −

x

u0 (t)dt = −

0

u1 (t)dt = −

0

u2 (t)dt = −

0

0

0

u3 (t)dt = −

x

0

x

x

(−t)dt =

x

1dt = −x,

0

x

x

0

1 2 x , 2!

(3.14)

1 1 2 t dt = − x3 , 2! 3!

x

−

1 3 1 t dt = − x4 , 3! 4!

and so on. Using (3.4) gives the series solution: 1 1 1 u(x) = 1 − x + x2 − x3 + x4 + · · · , 2! 3! 4! that converges to the closed form solution: u(x) = e−x .

(3.15) (3.16)

Example 3.2 Solve the following Volterra integral equation: x (t − x)u(t)dt. u(x) = 1 + 0

(3.17)

We notice that f (x) = 1, λ = 1, K(x, t) = t − x. Substituting the decomposition series (3.4) into both sides of (3.17) gives x ∞ ∞ un (x) = 1 + (t − x)un (t)dt, (3.18) n=0

or equivalently

0

n=0

3.2 Volterra Integral Equations of the Second Kind

u0 (x) + u1 (x) + u2 (x) + · · · = 1 +

69

x 0

(t − x) [u0 (t) + u1 (t) + u2 (t) + · · · ] dt. (3.19)

Proceeding as before we set the following recurrence relation: u0 (x) = 1, x uk+1 (x) = (t − x)uk (t)dt, k 0,

(3.20)

0

that gives u0 (x) = 1, x x 1 (t − x)u0 (t)dt = (t − x)dt = − x2 , u1 (x) = 2! 0 0 x 1 x 1 (t − x)u1 (t)dt = − (t − x)t2 dt = x4 , u2 (x) = 2! 0 4! 0 x 1 x 1 u3 (x) = (t − x)u2 (t)dt = (t − x)t4 dt = − x6 , 4! 0 6! 0 x x 1 1 (t − x)u3 (t)dt = − (t − x)t6 dt = x8 , u4 (x) = 6! 8! 0 0 and so on. The solution in a series form is given by 1 1 1 1 u(x) = 1 − x2 + x4 − x6 + x8 + · · · , 2! 4! 6! 8! and in a closed form by u(x) = cos x,

(3.21)

(3.22) (3.23)

obtained upon using the Taylor expansion for cos x. Example 3.3 Solve the following Volterra integral equation: x 1 u(x) = 1 − x − x2 − (t − x)u(t)dt. (3.24) 2 0 Notice that f (x) = 1 − x − 12 x2 , λ = −1, K(x, t) = t − x. Substituting the decomposition series (3.4) into both sides of (3.24) gives x ∞ ∞ 1 un(x) = 1 − x − x2 − (t − x)un (t)dt, (3.25) 2 0 n=0 n=0 or equivalently 1 u0 (x) + u1 (x) + u2 (x) + · · · = 1 − x − x2 − 2

0

x

(t − x) [u0 (t) + u1 (t) + · · · ] dt.

This allows us to set the following recurrence relation: 1 u0 (x) = 1 − x − x2 , 2 x (t − x)uk (t)dt, k 0, uk+1 (x) = − 0

(3.26)

(3.27)

70

3 Volterra Integral Equations

that gives 1 u0 (x) = 1 − x − x2 , 2 x 1 1 1 u1 (x) = − (t − x)u0 (t)dt = x2 − x3 − x4 , 2! 3! 4! 0 x 1 1 1 (t − x)u1 (t)dt = x4 − x5 − x6 , u2 (x) = − 4! 5! 6! 0 x 1 1 1 u3 (x) = − (t − x)u2 (t)dt = x6 − x7 − x8 , 6! 7! 8! 0 and so on. The solution in a series form is given by 1 1 1 u(x) = 1 − (x + x3 + x5 + x7 + · · · ), 3! 5! 7! and in a closed form by u(x) = 1 − sinh x,

(3.28)

(3.29) (3.30)

obtained upon using the Taylor expansion for sinh x. Example 3.4 We consider here the Volterra integral equation: x tu(t)dt. u(x) = 5x3 − x5 +

(3.31)

0

Identifying the zeroth component u0 (x) by the ﬁrst two terms that are not included under the integral sign, and using the ADM we set the recurrence relation as u0 (x) = 5x3 − x5 , x (3.32) tuk (t)dt, k 0. uk+1 (x) = 0

This in turn gives u0 (x) = 5x3 − x5 , x 1 tu0 (t)dt = x5 − x7 , u1 (x) = 7 0 x (3.33) 1 1 u2 (x) = tu1 (t)dt = x7 − x9 , 7 63 0 x 1 9 1 11 tu1 (t)dt = u3 (x) = x − x , 63 693 0 The solution in a series form is given by 3

1 7 1 9 1 11 1 7 1 5 5 u(x) = 5x − x + x − x + x − x + ( x9 − x )+· · · . 7 7 63 63 693 (3.34) We can easily notice the appearance of identical terms with opposite signs. Such terms are called noise terms that will be discussed later. Canceling the identical terms with opposite signs gives the exact solution

3.2 Volterra Integral Equations of the Second Kind

u(x) = 5x3 ,

71

(3.35)

that satisﬁes the Volterra integral equation (3.31). Example 3.5 We now consider the Volterra integral equation: x 1 1 u(t)dt. (3.36) u(x) = x + x4 + x2 + x5 − 2 5 0 Identifying the zeroth component u0 (x) by the ﬁrst four terms that are not included under the integral sign, and using the ADM we set the recurrence relation as 1 1 u0 (x) = x + x4 + x2 + x5 , 2 5 x (3.37) uk+1 (x) = − uk (t)dt, k 0. 0

This in turn gives 1 1 u0 (x) = x + x4 + x2 + x5 , 2 5 x 1 1 1 1 u1 (x) = − u0 (t)dt = − x2 − x5 − x3 − x6 , 2 5 6 30 0 x (3.38) 1 1 1 1 7 x , u1 (t)dt = x3 + x6 + x4 + u2 (x) = − 6 30 24 210 0 x 1 4 1 7 u2 (t)dt = − x − u3 (x) = − x + ··· . 24 210 0 The solution in a series form is given by 1 2 1 5 1 3 1 1 1 u(x) = x + x4 + x2 + x5 − x + x + x + x6 2 5 2 5 6 30 1 6 1 4 1 7 1 7 1 3 1 4 + x + x + x + x − x + x + ··· + ··· . 6 30 24 210 24 210 (3.39) We can easily notice the appearance of identical terms with opposite signs. This phenomenon of such terms is called noise terms phenomenon that will be presented later. Canceling the identical terms with opposite terms gives the exact solution u(x) = x + x4 . (3.40) Example 3.6 We ﬁnally solve the Volterra integral equation: 1 x 3 xt u(t)dt. u(x) = 2 + 3 0 Proceeding as before we set the recurrence relation

(3.41)

72

3 Volterra Integral Equations

u0 (x) = 2,

uk+1 (x) =

1 3

x 0

xt3 uk (t)dt,

k 0.

(3.42)

This in turn gives u0 (x) = 2, 1 x 3 1 u1 (x) = xt u0 (t)dt = x5 , 3 0 6 x 1 1 10 x , xt3 u1 (t)dt = u2 (x) = 3 0 162 (3.43) 1 x 3 1 15 xt u2 (t)dt = u3 (x) = x , 3 0 6804 1 x 3 1 u4 (x) = x20 , xt u3 (t)dt = 3 0 387828 and so on. The solution in a series form is given by 1 1 1 1 x15 + x20 + · · · . (3.44) x10 + u(x) = 2 + x5 + 3 4 5 6 6·3 6 · 3 · 14 6 · 3 · 14 · 19 It seems that an exact solution is not obtainable. The obtained series solution can be used for numerical purposes. The more components that we determine the higher accuracy level that we can achieve.

Exercises 3.2.1 In Exercises 1–26, solve the following Volterra integral equations by using the Adomian decomposition method: x x u(t)dt 2. u(x) = 6x − x3 + (x − t)u(t)dt 1. u(x) = 6x − 3x2 + 0

x 1 3. u(x) = 1 − x2 + u(t)dt 2 0 x 5. u(x) = 1 + x + (x − t)u(t)dt 0 x

0

x 2 4. u(x) = x − x3 − 2 u(t)dt 3 0 x 6. u(x) = 1 − x + (x − t)u(t)dt

7. u(x) = 1 + x − 9. u(x) = 1 −

x 0

11. u(x) = x − 13. u(x) = 1 +

0

0 x

8. u(x) = 1 − x − 10. u(x) = 1 +

0

12. u(x) = x +

u(t)dt

0

17. u(x) = 1 − x2 −

14. u(x) = 1 −

0 x

tu(t)dt

x

0

(x − t)u(t)dt

16. u(x) = 1 − 2

(x − t)u(t)dt

(x − t)u(t)dt

x

(x − t)u(t)dt u(t)dt

0

x

0

x

(x − t)u(t)dt

0

15. u(x) = 1 + 2

(x − t)u(t)dt

(x − t)u(t)dt x

0 x

x 0

tu(t)dt

18. u(x) = −2 + 3x − x2 −

x

0

(x − t)u(t)dt

3.2 Volterra Integral Equations of the Second Kind x 19. u(x) = x2 + (x − t)u(t)dt 20. u(x) = −2 + 2x + x2 + 0

21. u(x) = 1 + 2x + 4

x

0

(x − t)u(t)dt

22. u(x) = 5 + 2x2 −

73 x

0

x 0

(x − t)u(t)dt

(x − t)u(t)dt

1 2 1 x x + (x − t)2 u(t)dt 2 2 0 1 x 1 (x − t)3 u(t)dt 24. u(x) = 1 − x2 + 2 6 0 1 1 x 25. u(x) = 1 + x + (x − t + 1)u(t)dt 2 2 0 x 26. u(x) = 1 + x2 − (x − t + 1)2 u(t)dt 23. u(x) = 1 + x +

0

In Exercises 27–30, use the Adomian decomposition method to ﬁnd the series solution 1 x 2 1 x 27. u(x) = 3 + xt u(t)dt 28. u(x) = 3 + (x + t2 )u(t)dt 4 0 4 0 1 x 2 1 x 2 29. u(x) = 1 + (x − t2 )u(t)dt 30. u(x) = 1 + x u(t)dt 2 0 2 0

3.2.2 The Modiﬁed Decomposition Method As shown before, the Adomian decomposition method provides the solution in an inﬁnite series of components. The components uj , j 0 are easily computed if the inhomogeneous term f (x) in the Volterra integral equation: x K(x, t)u(t)dt (3.45) u(x) = f (x) + λ 0

consists of a polynomial. However, if the function f (x) consists of a combination of two or more of polynomials, trigonometric functions, hyperbolic functions, and others, the evaluation of the components uj , j 0 requires cumbersome work. A reliable modiﬁcation of the Adomian decomposition method was developed by Wazwaz and presented in [7–9]. The modiﬁed decomposition method will facilitate the computational process and further accelerate the convergence of the series solution. The modiﬁed decomposition method will be applied, wherever it is appropriate, to all integral equations and diﬀerential equations of any order. It is interesting to note that the modiﬁed decomposition method depends mainly on splitting the function f (x) into two parts, therefore it cannot be used if the function f (x) consists of only one term. The modiﬁed decomposition method will be outlined and employed in this section and in other chapters as well. To give a clear description of the technique, we recall that the standard Adomian decomposition method admits the use of the recurrence relation:

74

3 Volterra Integral Equations

u0 (x) = f (x), x K(x, t)uk (t)dt, k 0, uk+1 (x) = λ

(3.46)

0

where the solution u(x) is expressed by an inﬁnite sum of components deﬁned before by ∞ un (x). (3.47) u(x) = n=0

In view of (3.46), the components un (x), n 0 can be easily evaluated. The modiﬁed decomposition method [7–9] introduces a slight variation to the recurrence relation (3.46) that will lead to the determination of the components of u(x) in an easier and faster manner. For many cases, the function f (x) can be set as the sum of two partial functions, namely f1 (x) and f2 (x). In other words, we can set (3.48) f (x) = f1 (x) + f2 (x). In view of (3.48), we introduce a qualitative change in the formation of the recurrence relation (3.46). To minimize the size of calculations, we identify the zeroth component u0 (x) by one part of f (x), namely f1 (x) or f2 (x). The other part of f (x) can be added to the component u1 (x) among other terms. In other words, the modiﬁed decomposition method introduces the modiﬁed recurrence relation: u0 (x) = f1 (x), x K(x, t)u0 (t)dt, u1 (x) = f2 (x) + λ (3.49) 0 x uk+1 (x) = λ K(x, t)uk (t)dt, k 1. 0

This shows that the diﬀerence between the standard recurrence relation (3.46) and the modiﬁed recurrence relation (3.49) rests only in the formation of the ﬁrst two components u0 (x) and u1 (x) only. The other components uj , j 2 remain the same in the two recurrence relations. Although this variation in the formation of u0 (x) and u1 (x) is slight, however it plays a major role in accelerating the convergence of the solution and in minimizing the size of computational work. Moreover, reducing the number of terms in f1 (x) aﬀects not only the component u1 (x), but also the other components as well. This result was conﬁrmed by several research works. Two important remarks related to the modiﬁed method [7–9] can be made here. First, by proper selection of the functions f1 (x) and f2 (x), the exact solution u(x) may be obtained by using very few iterations, and sometimes by evaluating only two components. The success of this modiﬁcation depends only on the proper choice of f1 (x) and f2 (x), and this can be made through trials only. A rule that may help for the proper choice of f1 (x) and f2 (x) could not be found yet. Second, if f (x) consists of one term only, the standard decomposition method can be used in this case.

3.2 Volterra Integral Equations of the Second Kind

75

It is worth mentioning that the modiﬁed decomposition method will be used for Volterra and Fredholm integral equations, linear and nonlinear equations. The modiﬁed decomposition method will be illustrated by discussing the following examples. Example 3.7 Solve the Volterra integral equation by using the modiﬁed decomposition method: x u(x) = sin x + (e − ecos x ) − ecos t u(t)dt. (3.50) 0

We ﬁrst split f (x) given by f (x) = sin x + (e − ecos x ),

(3.51)

f1 (x) = sin x, f2 (x) = e − ecos x .

(3.52)

into two parts, namely

We next use the modiﬁed recurrence formula (3.49) to obtain u0 (x) = f1 (x) = sin x, x cos x u1 (x) = (e − e )− ecos t u0 (t)dt = 0, 0

uk+1 (x) = −

(3.53)

x

K(x, t)uk (t)dt = 0,

0

k 1.

It is obvious that each component of uj , j 1 is zero. This in turn gives the exact solution by u(x) = sin x. (3.54) Example 3.8 Solve the Volterra integral equation by using the modiﬁed decomposition method: x π sec x (3.55) u(x) = sec x tan x + (e − e) − esec t u(t)dt, x < . 2 0 Proceeding as before we split f (x) into two parts f1 (x) = sec x tan x,

f2 (x) = esec x − e.

We next use the modiﬁed recurrence formula (3.49) to obtain u0 (x) = f1 (x) = sec x tan x, x esec t u0 (t)dt = 0, u1 (x) = (esec x − e) − 0

uk+1 (x) = −

(3.56)

(3.57)

x 0

K(x, t)uk (t)dt = 0,

k 1.

It is obvious that each component of uj , j 1 is zero. This in turn gives the exact solution by

76

3 Volterra Integral Equations

u(x) = sec x tan x.

(3.58)

Example 3.9 Solve the Volterra integral equation by using the modiﬁed decomposition method: x u(x) = 2x + sin x + x2 − cos x + 1 − u(t)dt. (3.59) 0

The function f (x) consists of ﬁve terms. By trial we divide f (x) given by f (x) = 2x + sin x + x2 − cos x + 1, into two parts, ﬁrst two terms and the next three terms to ﬁnd f1 (x) = 2x + sin x, f2 (x) = x2 − cos x + 1. We next use the modiﬁed recurrence formula (3.49) to obtain u0 (x) = 2x + sin x, x u0 (t)dt = 0, u1 (x) = x2 − cos x + 1 − 0

uk+1 (x) = −

(3.60)

(3.61)

(3.62)

x 0

K(x, t)uk (t)dt = 0,

k 1.

It is obvious that each component of uj , j 1 is zero. The exact solution is given by u(x) = 2x + sin x. (3.63) Example 3.10 Solve the Volterra integral equation by using the modiﬁed decomposition method: x 1 u(x) = 1 + x2 + cos x − x − x3 − sin x + u(t)dt. (3.64) 3 0 The function f (x) consists of six terms. By trial we split f (x) given by 1 (3.65) f (x) = 1 + x2 + cos x − x − x3 − sin x, 3 into two parts, the ﬁrst three terms and the next three terms, hence we set f1 (x) = 1 + x2 + cos x, (3.66) 1 f2 (x) = −(x + x3 + sin x). 3 Using the modiﬁed recurrence formula (3.49) gives u0 (x) = 1 + x2 + cos x, x 1 3 u1 (x) = −(x + x + sin x) + u0 (t)dt = 0, 3 (3.67) 0 x K(x, t)uk (t)dt = 0, k 1. uk+1 (x) = 0

3.2 Volterra Integral Equations of the Second Kind

77

As a result, the exact solution is given by u(x) = 1 + x2 + cos x.

(3.68)

Exercises 3.2.2 Use the modiﬁed decomposition method to solve the following Volterra integral equations: x u(t)dt 1. u(x) = cos x + sin x − 0

2. u(x) = sinh x + cosh x − 1 − 3. u(x) = 2x = 3x2 + (ex

2

+x3

3

4. u(x) = 3x2 + (1 − e−x ) − 2

5. u(x) = 2x − (1 − e−x ) +

x

0

u(t)dt

− 1) −

x

0 x 0

3

2

et

0

e−x

e−x

x

+t3

+t2

2

+t3

u(t)dt

u(t)dt

u(t)dt

x 2 x xtu(t)dt (1 − e−x ) − 2 0 x xu(t)dt 7. u(x) = cosh x + x sinh x − 2

6. u(x) = e−x −

x

x

8. u(x) = e + xe − x −

0

x

0

xu(t)dt

9. u(x) = 1 + sin x + x + x2 − x cos x − 10. u(x) = ex − xex + sin x + x cos x +

x

xu(t)dt

0

x

xu(t)dt

0

x 1 3 xu(t)dt x + cosh x + x sinh x − 2 0 x

esin t u(t)dt 12. u(x) = cos x − 1 − esin x x − x

11. u(x) = 1 + x + x2 +

0

13. u(x) = sec x2 − 1 − etan x x − x 14. u(x) = cosh x +

x

0

x x 1 − esinh x + 2 2

1 1 e − ecosh x + 10 10 x 1 tu(t)dt 16. u(x) = x3 − x5 + 5 10 0

15. u(x) = sinh x +

etan t u(t)dt x

0

esinh t u(t)dt x

0

ecosh t u(t)dt

78

3 Volterra Integral Equations

3.2.3 The Noise Terms Phenomenon It was shown before that the modiﬁed decomposition method presents a reliable tool for accelerating the computational work. However, a proper selection of f1 (x) and f2 (x) is essential for a successful use of this technique. A useful tool that will accelerate the convergence of the Adomian decomposition method is developed. The new technique depends mainly on the so-called noise terms phenomenon that demonstrates a fast convergence of the solution. The noise terms phenomenon can be used for all diﬀerential and integral equations. The noise terms, if existed between the components u0 (x) and u1 (x), will provide the exact solution by using only the ﬁrst two iterations. In what follows, we outline the main concepts of the noise terms : 1. The noise terms are deﬁned as the identical terms with opposite signs that arise in the components u0 (x) and u1 (x). Other noise terms may appear between other components. As stated above, these identical terms with opposite signs may exist for some equations, and may not appear for other equations. 2. By canceling the noise terms between u0 (x) and u1 (x), even though u1 (x) contains further terms, the remaining non-canceled terms of u0 (x) may give the exact solution of the integral equation. The appearance of the noise terms between u0 (x) and u1 (x) is not always suﬃcient to obtain the exact solution by canceling these noise terms. Therefore, it is necessary to show that the non-canceled terms of u0 (x) satisfy the given integral equation. On the other hand, if the non-canceled terms of u0 (x) did not satisfy the given integral equation, or the noise terms did not appear between u0 (x) and u1 (x), then it is necessary to determine more components of u(x) to determine the solution in a series form as presented before. 3.It was formally shown that the noise terms appear for speciﬁc cases of inhomogeneous diﬀerential and integral equations, whereas homogeneous equations do not give rise to noise terms. The conclusion about the selfcanceling noise terms was based on solving several speciﬁc diﬀerential and integral models. However, a proof for this conclusion was not given. For further readings about the noise terms phenomenon, see [7,10]. 4. It was formally proved in [7,10] that the appearance of the noise terms is governed by a necessary condition. The conclusion made in [7,10] is that the zeroth component u0 (x) must contain the exact solution u(x) among other terms. In addition, it was proved that the inhomogeneity condition of the equation does not always guarantee the appearance of the noise terms as examined in [10]. A useful summary about the noise terms phenomenon can be drawn as follows:

3.2 Volterra Integral Equations of the Second Kind

79

1. The noise terms are deﬁned as the identical terms with opposite signs that may appear in the components u0 (x) and u1 (x) and in the other components as well. 2. The noise terms appear only for speciﬁc types of inhomogeneous equations whereas noise terms do not appear for homogeneous equations. 3. Noise terms may appear if the exact solution of the equation is part of the zeroth component u0 (x). 4. Veriﬁcation that the remaining non-canceled terms satisfy the integral equation is necessary and essential. The phenomenon of the useful noise terms will be explained by the following illustrative examples. Example 3.11 Solve the Volterra integral equation by using noise terms phenomenon: 3 x tu(t)dt. (3.69) u(x) = 8x + x3 − 8 0 Following the standard Adomian method we set the recurrence relation: u0 (x) = 8x + x3 , 3 x (3.70) uk+1 (x) = − tuk (t)dt, k 0. 8 0 This gives u0 (x) = 8x + x3 , (3.71) 3 x 3 tu0 (t)dt = − x5 − x3 . u1 (x) = − 8 0 40 The noise terms ±x3 appear in u0 (x) and u1 (x). Canceling this term from the zeroth component u0 (x) gives the exact solution: u(x) = 8x, (3.72) that satisﬁes the integral equation. Notice that if the modiﬁed method is used, we select u0 (x) = 8x. As a result, we ﬁnd that u1 (x) = 0. This in turn gives the same result. Example 3.12 Solve the Volterra integral equation by using noise terms phenomenon: x 1 (x − t)2 u(t)dt. (3.73) u(x) = −2 + x + x2 + x4 + sin x + 2 cos x − 12 0 Following the standard Adomian method we set the recurrence relation: 1 u0 (x) = −2 + x + x2 + x4 + sin x + 2 cos x, 12 x (3.74) 2 uk+1 (x) = − (x − t) uk (t)dt, k 0. 0

This gives

80

3 Volterra Integral Equations

1 u0 (x) = −2 + x + x2 + x4 + sin x + 2 cos x, 12 3 x tu0 (t)dt u1 (x) = − 8 0 1 1 2 1 7 = 2 − x2 − x4 − 2 cos x + 4 sin x − x5 + x3 − x − 4x. 12 30 3 1260 (3.75) 1 4 x and ±2 cos x appear in u0 (x) and u1 (x). The noise terms ∓2, ±x2 , ± 12 Canceling these terms from the zeroth component u0 (x) gives the exact solution u(x) = x + sin x, (3.76) that satisﬁes the integral equation. It is to be noted that the other terms of u1 (x) vanish in the limit with other terms of the other components. Example 3.13 Solve the Volterra integral equation by using noise terms phenomenon x 1 1 2 u(t)dt. (3.77) u(x) = x − sinh(2x) + sinh x + 2 4 0 We set the recurrence relation: 1 1 u0 (x) = x − sinh(2x) + sinh2 x, 2 4 x (3.78) uk (t)dt, k 0. uk+1 (x) = 0

This gives 1 1 x − sinh(2x) + sinh2 x, 2 4 x 1 1 u1 (x) = u0 (t)dt = − x + sinh(2x) + 2 4 0 1 1 The noise terms ± 2 x and ∓ 4 sinh(2x) appear these terms from the zeroth component u0 (x) u0 (x) =

(3.79) 1 1 2 1 + x − cosh2 x. 4 4 4 in u0 (x) and u1 (x). Canceling gives the exact solution:

u(x) = sinh2 x, that satisﬁes the integral equation.

(3.80)

Example 3.14 Show that the exact solution for the Volterra integral equation: x 1 u(t)dt, u(x) = −1 + x + x2 + 2ex − 2 0 cannot be obtained by using the noise terms phenomenon. We set the recurrence relation: 1 u0 (x) = −1 + x + x2 + 2ex , 2 x uk (t)dt, k 0. uk+1 (x) = − 0

(3.81)

(3.82)

3.2 Volterra Integral Equations of the Second Kind

81

This gives 1 u0 (x) = −1 + x + x2 + 2ex , 2 x (3.83) 1 1 u1 (x) = − u0 (t)dt = − x2 − 2ex + 2 + x − x3 . 2 6 0 1 2 x The noise terms ± 2 x and ±2e appear in u0 (x) and u1 (x). Canceling these terms from the zeroth component u0 (x) gives u ˜(x) = x − 1, (3.84) that does not satisfy the integral equation. This conﬁrms our belief that the non-canceled terms of u0 (x) do not always give the exact solution, and therefore justiﬁcation is necessary. The exact solution is given by u(x) = x + ex ,

(3.85)

that can be easily obtained by using the modiﬁed decomposition method by setting f1 (x) = x + ex . Exercises 3.2.3 Use the noise terms phenomenon to solve the following Volterra integral equations: x x 1. u(x) = 6x + 3x2 − u(t)dt 2. u(x) = 6x + 3x2 − xu(t)dt 0

3. u(x) = 6x + 2x3 −

x

0

0

tu(t)dt 4. u(x) = x + x2 − 2x3 − x4 + 12

5. u(x) = −2 + x2 + sin x + 2 cos x − 6. u(x) = 2x − 2 sin x + cos x −

x

0

x 0

(x − t)2 u(t)dt

(x − t)2 u(t)dt

7. u(x) = sinh x + x sinh x − x2 cosh x +

x

xtu(t)dt

0

8. u(x) = x + cosh x + x2 sinh x − x cosh x − 9. u(x) = sec2 x − tan x +

x

0

xtu(t)dt

x

0

u(t)dt

1 1 10. u(x) = − x − sin(2x) + cos2 x + 2 4

x 0

u(t)dt

x 1 1 u(t)dt 11. u(x) = − x + sin(2x) + sin2 x + 2 4 0 x 12. u(x) = −x + tan x + tan2 x − u(t)dt 0

x

0

(x − t)u(t)dt

82

3 Volterra Integral Equations

3.2.4 The Variational Iteration Method In this section we will study the newly developed variational iteration method that proved to be eﬀective and reliable for analytic and numerical purposes. The variational iteration method (VIM) established by Ji-Huan He [11–12] is now used to handle a wide variety of linear and nonlinear, homogeneous and inhomogeneous equations. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the so called Adomian polynomials that we need for nonlinear problems. Moreover, the method gives the solution in a series form that converges to the closed form solution if an exact solution exists. The obtained series can be employed for numerical purposes if exact solution is not obtainable. In what follows, we present the main steps of the method. Consider the diﬀerential equation: Lu + N u = g(t), (3.86) where L and N are linear and nonlinear operators respectively, and g(t) is the source inhomogeneous term. The variational iteration method presents a correction functional for equation (3.86) in the form: x un+1 (x) = un (x) + λ(ξ) (Lun (ξ) + N u ˜n (ξ) − g(ξ)) dξ, (3.87) 0

where λ is a general Lagrange’s multiplier, noting that in this method λ may be a constant or a function, and u ˜n is a restricted value that means it behaves as a constant, hence δ˜ un = 0, where δ is the variational derivative. The Lagrange multiplier λ can be identiﬁed optimally via the variational theory as will be seen later. For a complete use of the variational iteration method, we should follow two steps, namely: 1. the determination of the Lagrange multiplier λ(ξ) that will be identiﬁed optimally, and 2. with λ determined, we substitute the result into (3.87) where the restrictions should be omitted. Taking the variation of (3.87) with respect to the independent variable un we ﬁnd x δun+1 δ =1+ λ(ξ) (Lun (ξ) + N u ˜n (ξ) − g(ξ)) dξ , (3.88) δun δun 0 or equivalently x λ(ξ) (Lun (ξ)) dξ . (3.89) δun+1 = δun + δ 0

3.2 Volterra Integral Equations of the Second Kind

83

Integration by parts is usually used for the determination of the Lagrange multiplier λ(ξ). In other words we can use x x λ(ξ)un (ξ)dξ = λ(ξ)un (ξ) − λ (ξ)un (ξ)dξ, 0

0

x

0 x 0

λ(ξ)un (ξ)dξ = λ(ξ)un (ξ) − λ (ξ)un (ξ) + λ(ξ)u n (ξ)dξ

0

x

x

0

λ (ξ)un (ξ)dξ,

= λ(ξ)un (ξ) − λ (ξ)un (ξ) + λ (ξ)un (ξ) −

x

(3.90)

λ (ξ)un (ξ)dξ,

0

λ(ξ)u(iv) n (ξ)dξ = λ(ξ)un (ξ) − λ (ξ)un (ξ) + λ (ξ)un (ξ) − λ un (ξ)

x

+ 0

λ(iv) (ξ)un (ξ)dξ,

and so on. These identities are obtained by integrating by parts. For example, if Lun (ξ) = un (ξ) in (3.89), then (3.89) becomes x δun+1 = δun + δ λ(ξ) (Lun (ξ)) dξ .

(3.91)

Integrating the integral of (3.91) by parts using (3.90) we obtain x λ (ξ)δun (ξ)dξ, δun+1 = δun + δλ(ξ)un (ξ) −

(3.92)

0

0

or equivalently

δun+1 = δun (ξ)(1 + λ|ξ=x ) −

x 0

λ δun dξ.

(3.93)

The extremum condition of un+1 requires that δun+1 = 0. This means that the left hand side of (3.93) is zero, and as a result the right hand side should be 0 as well. This yields the stationary conditions: 1 + λ|ξ=x = 0,

λ |ξ=x = 0.

(3.94)

This in turn gives λ = −1.

(3.95)

As a second example, if Lun (ξ) = un (ξ) in (3.89), then (3.89) becomes x λ(ξ) (Lun (ξ)) dξ . (3.96) δun+1 = δun + δ 0

Integrating the integral of (3.96) by parts using (3.90) we obtain x λ δun dξ, δun+1 = δun + δλ((un ) )x0 − (λ δun)x0 +

(3.97)

0

or equivalently δun+1 = δun (ξ)(1 − λ |ξ=x ) + δλ((un ) |ξ=x ) +

x 0

λ δun dξ,

(3.98)

84

3 Volterra Integral Equations

The extremum condition of un+1 requires that δun+1 = 0. This means that the left hand side of (3.98) is zero, and as a result the right hand side should be 0 as well. This yields the stationary conditions: 1 − λ |ξ=x = 0, λ|ξ=x = 0, λ |ξ=x = 0. (3.99) This in turn gives λ = ξ − x.

(3.100)

Having determined the Lagrange multiplier λ(ξ), the successive approximations un+1 , n 0, of the solution u(x) will be readily obtained upon using selective function u0 (x). However, for fast convergence, the function u0 (x) should be selected by using the initial conditions as follows: u0 (x) = u(0), for ﬁrst order un , u0 (x) = u(0) + xu (0), for second order un , (3.101) 1 u0 (x) = u(0) + xu (0) + x2 u (0), for third order u n, 2! .. . and so on. Consequently, the solution u(x) = lim un (x). n→∞

(3.102)

In other words, the correction functional (3.87) will give several approximations, and therefore the exact solution is obtained as the limit of the resulting successive approximations. The determination of the Lagrange multiplier plays a major role in the determination of the solution of the problem. In what follows, we summarize some iteration formulae that show ODE, its corresponding Lagrange multipliers, and its correction functional respectively:

⎧ ⎪ ⎨ u + f (u(ξ), u (ξ)) = 0, λ = −1, x (i) ⎪ [un + f (un , un )] dξ, ⎩ un+1 = un − 0

⎧ ⎪ ⎨ u + f (u(ξ), u (ξ), u (ξ)) = 0, λ = (ξ − x), x (ii) ⎪ (ξ − x) [u ⎩ un+1 = un + n + f (un , un , un )] dξ, 0

⎧ 1 2 ⎪ ⎪ ⎨ u + f (u(ξ), u (ξ), u (ξ), u (ξ)) = 0, λ = − 2! (ξ − x) , x (iii) 1 ⎪ ⎪ ⎩ un+1 = un − (ξ − x)2 [u n + f (un , . . . , un )] dξ, 2! 0 ⎧ 1 (iv) ⎪ ⎪ + f (u(ξ), u (ξ), u (ξ), u (ξ), u(iv) (ξ)) = 0, λ = (ξ − x)3 , ⎨u 3! x (iv) 1 ⎪ (iv) ⎪ ⎩ un+1 = un + (ξ − x)3 u n + f (un , un , . . . , un ) dξ, 0 3!

3.2 Volterra Integral Equations of the Second Kind

85

and generally ⎧ 1 ⎪ ⎪ (ξ − x)(n−1) , ⎨ u(n) + f (u(ξ), u (ξ), . . . , u(n) (ξ)) = 0, λ = (−1)n (n − 1)! (v) x 1 ⎪ (n) n ⎪ (ξ − x)(n−1) u ⎩ un+1 = un + (−1) n + f (un , . . . , un ) dξ, (n − 1)! 0 for n 1.

To use the variational iteration method for solving Volterra integral equations, it is necessary to convert the integral equation to an equivalent initial value problem or to an equivalent integro-diﬀerential equation. As deﬁned before, integro-diﬀerential equation is an equation that contains diﬀerential and integral operators in the same equation. The integro-diﬀerential equations will be studied in details in Chapter 5. The conversion process is presented in Section 2.5.1. However, for comparison reasons, we will examine the obtained initial value problem by two methods, namely, standard methods used for solving ODEs, and by using the variational iteration method as will be seen by the following examples. Example 3.15 Solve the Volterra integral equation by using the variational iteration method x u(t)dt. (3.103) u(x) = 1 + 0

Using Leibnitz rule to diﬀerentiate both sides of (3.103) gives u (x) − u(x) = 0.

(3.104)

Substituting x = 0 into (3.103) gives the initial condition u(0) = 1. Using the variational iteration method The correction functional for equation (3.104) is x un+1(x) = un (x) + λ(ξ)(un (ξ) − u ˜n (ξ))dξ.

(3.105)

Using the formula (i) given above leads to λ = −1.

(3.106)

0

Substituting this value of the Lagrange multiplier λ = −1 into the functional (3.105) gives the iteration formula: x (un (ξ) − un (ξ)) dξ. (3.107) un+1 (x) = un(x) − 0

As stated before, we can use the initial condition to select u0 (x) = u(0) = 1. Using this selection into (3.105) gives the following successive approximations: u0 (x) = 1, x (u0 (ξ) − u0 (ξ)) dξ = 1 + x, u1 (x) = 1 − 0

u2 (x) = 1 + x −

0

x

(u1 (ξ) − u1 (ξ)) dξ = 1 + x +

1 2 x , 2!

86

3 Volterra Integral Equations

u3 (x) = 1 + x +

1 2 x − 2!

x 0

(u2 (ξ) − u2 (ξ)) dξ = 1 + x +

1 2 1 x + x3 , 2! 3! (3.108)

and so on. The VIM admits the use of u(x) = limn→∞ un (x) 1 2 1 1 1 x + x3 + x4 + · · · + xn ), 2! 3! 4! n! that gives the exact solution by

(3.109)

= limn→∞ (1 + x +

u(x) = ex .

(3.110)

Using ODEs method The ODE (3.104) is of ﬁrst order, therefore the integrating factor μ(x) is given by x μ(x) = e (−1)dx = e−x . (3.111) For ﬁrst order ODE, we use the formula: x 1 μq(x)dx + C = Cex . (3.112) u(x) = μ To obtain the particular solution, we use the initial condition u(0) = 1 to ﬁnd that C = 1. This gives the particular solution: u(x) = ex .

(3.113)

Example 3.16 Solve the Volterra integral equation by using the variational iteration method x (x − t) u(t)dt. (3.114) u(x) = x + 0

Using Leibnitz rule to diﬀerentiate both sides of (3.114) once with respect to x gives the integro-diﬀerential equation: x u (x) = 1 + u(t)dt, u(0) = 0. (3.115) 0

However, by diﬀerentiating (3.115) with respect to x we obtain the diﬀerential equation: (3.116) u (x) = u(x). Substituting x = 0 into (3.114) and (3.115) gives the initial conditions u(0) = 0 and u (0) = 1. The resulting initial value problem, that consists of a second order ODE and initial conditions is given by (3.117) u (x) − u(x) = 0, u(0) = 0, u (0) = 1. The integro-diﬀerential equation (3.115) and the initial value problem (3.116) will be handled independently by using the variational iteration method. Using the variational iteration method (i) We ﬁrst start using the variational iteration method to handle the integrodiﬀerential equation (3.115) given by

3.2 Volterra Integral Equations of the Second Kind

u (x) = 1 +

87

x

u(t)dt, u(0) = 0.

(3.118)

0

The correction functional for Eq. (3.118) is x λ(ξ) un (ξ) − 1 − un+1 (x) = un (x) + 0

ξ 0

u ˜n (r)dr

dξ.

(3.119)

Using the formula (i) for λ we ﬁnd that λ = −1. (3.120) Substituting this value of the Lagrange multiplier λ = −1 into the functional (3.119) gives the iteration formula: x ξ un (ξ) − 1 − un+1 (x) = un (x) − un (r)dr dξ. (3.121) 0

0

We can use the initial conditions to select u0 (x) = u(0) = 0. Using this selection into (3.121) gives the following successive approximations: u0 (x) = 0, x ξ u1 (x) = x − u0 (r)dr dξ = x, u0 (ξ) − 1 − 0

u2 (x) = x −

0

u3 (x) = x −

x

0

x

0

u1 (ξ)

−1−

u2 (ξ)

−1−

u1 (r)dr

0

0

ξ

dξ = x +

ξ

u2 (r)dr

1 3 x , 3!

(3.122)

1 1 dξ = x + x3 + x5 , 3! 5!

.. . 1 3 1 1 1 x + x5 + x 7 + · · · + x2n+1 . 3! 5! 7! (2n + 1)! The VIM admits the use of u(x) = lim un (x), un (x) = x +

n→∞

(3.123)

that gives the exact solution by u(x) = sinh x. (3.124) (ii) We can obtain the same result by applying the variational iteration method to handle the initial value problem (3.117) given by (3.125) u (x) − u(x) = 0, u(0) = 0, u (0) = 1. The correction functional for Eq. (3.117) is x un+1 (x) = un (x) + λ(ξ) (un (ξ) − u ˜n (ξ)) dξ.

(3.126)

0

Using the formula (ii) given above leads to λ = ξ − x.

(3.127)

88

3 Volterra Integral Equations

Substituting this value of the Lagrange multiplier λ = ξ−x into the functional (3.126) gives the iteration formula: x un+1 (x) = un (x) + (ξ − x) (un (ξ) − un (ξ)) dξ. (3.128) 0

We can use the initial conditions to select u0 (x) = u(0) + xu (0) = x. Using this selection into (3.128) gives the following successive approximations u0 (x) = x, x 1 (ξ − x) (u0 (ξ) − u0 (ξ)) dξ = x + x3 , u1 (x) = x + 3! 0 x 1 1 1 (ξ − x) (u1 (ξ) − u1 (ξ)) dξ = x + x3 + x5 , u2 (x) = x + x3 + 3! 3! 5! 0 x 1 1 u3 (x) = x + x3 + x5 + (ξ − x) (u2 (ξ) − u2 (ξ)) dξ 3! 5! 0 1 5 1 7 1 3 = x+ x + x + x , 3! 5! 7! .. . 1 1 1 1 un (x) = x + x3 + x5 + x7 + · · · + x2n+1 . 3! 5! 7! (2n + 1)! (3.129) The VIM admits the use of u(x) = lim un (x), (3.130) n→∞

that gives the exact solution by u(x) = sinh x. (3.131) Standard methods for solving ODEs The initial value problem (3.125) is of second order, therefore the auxiliary equation is of the form (3.132) r2 − 1 = 0, that gives r = ±1. This in turn gives the general solution by u(x) = A sinh x + B cosh x.

(3.133)

To obtain the particular solution, we use the initial conditions u(0) = 0, u (0) = 1 to ﬁnd that the particular solution is given by u(x) = sinh x.

(3.134)

Example 3.17 Solve the Volterra integral equation by using the variational iteration method x 1 (x − t)u(t)dt. (3.135) u(x) = 1 + x + x3 − 3! 0 Using Leibnitz rule to diﬀerentiate both sides of (3.135) once with respect to x gives the integro-diﬀerential equation:

3.2 Volterra Integral Equations of the Second Kind x 1 2 u(t)dt, u(0) = 1, x − 2! 0 and by diﬀerentiating again we obtain the initial value problem

u (x) = 1 +

89

u (x) + u(x) = x, u(0) = 1, u (0) = 1.

(3.136)

(3.137)

Using the variational iteration method (i) We ﬁrst start using the variational iteration method to handle the integrodiﬀerential equation (3.136) given by x 1 u(t)dt, u(0) = 1, (3.138) u (x) = 1 + x2 − 2! 0 The correction functional for Eq. (3.138) is x ξ 1 2 λ(ξ) un (ξ) − 1 − ξ + u ˜n (r)dr dξ. (3.139) un+1 (x) = un (x) − 2 0 0 Using the formula (i) for λ we ﬁnd that λ = −1.

(3.140)

Substituting this value of the Lagrange multiplier λ = −1 into the functional (3.139) gives the iteration formula x ξ 1 2 un (r)dr dξ. (3.141) un (ξ) − 1 − ξ + un+1 (x) = un (x) − 2 0 0 We can use the initial conditions to select u0 (x) = u(0) = 1. Using this selection into (3.141) gives the following successive approximations u0 (x) = 1, x ξ 1 2 u0 (r)dr dξ u0 (ξ) − 1 − ξ + u1 (x) = x − 2 0 0 1 1 = 1 + x − x2 + x 3 , 2! 3! x ξ 1 u2 (x) = x − u1 (r)dr dξ u1 (ξ) − 1 − ξ 2 + 2 0 0 1 1 1 (3.142) = 1 + x − x2 + x 4 − x5 , 2! 4! 5! x ξ 1 u2 (ξ) − 1 − ξ 2 + u3 (x) = x − u2 (r)dr dξ 2 0 0 1 1 1 = 1 + x − x2 + x 4 − x6 , 2! 4! 6! .. . 1 1 1 (−1)n 2n x . un (x) = x + 1 − x2 + x4 − x6 + · · · + 2! 4! 6! (2n)! The VIM admits the use of

90

3 Volterra Integral Equations

u(x) = lim un (x), n→∞

(3.143)

that gives the exact solution by u(x) = x + cos x. (3.144) (ii) We next use the variational iteration method for solving the initial value problem u (x) + u(x) = x, u(0) = 1, u (0) = 1. (3.145) The correction functional for Eq. (3.145) is x λ(ξ) (un (ξ) + u ˜n (ξ) − ξ) dξ. un+1 (x) = un (x) +

(3.146)

0

Using the formula (ii) given above leads to λ = ξ − x.

(3.147)

Substituting this value of the Lagrange multiplier λ = ξ−x into the functional (3.146) gives the iteration formula x (ξ − x) (un (ξ) + un (ξ) − ξ) dξ. (3.148) un+1 (x) = un (x) + 0

We can use the initial conditions to select u0 (x) = u(0) + xu (0) = 1 + x. Using this selection into (3.148) gives the following successive approximations u0 (x) = 1 + x, x (ξ − x) (u0 (ξ) + u0 (ξ) − ξ) dξ u1 (x) = 1 + x + 0

1 = 1 + x − x2 , 2!

x 1 2 x + (ξ − x) (u1 (ξ) + u1 (ξ) − ξ) dξ 2! 0 1 4 1 2 = 1+x− x + x , (3.149) 2! 4! x 1 2 1 4 u3 (x) = 1 + x − x + x + (ξ − x) (u2 (ξ) + u2 (ξ) − ξ) dξ 2! 4! 0 1 1 1 = 1 + x − x2 + x4 − x6 2! 4! 6! .. . 1 1 1 1 (−1)n 2n . x un (x) = x + 1 − x2 + x4 − x6 + x8 − · · · + 2! 4! 6! 8! (2n)! Proceeding as before, the VIM gives the exact solution by u(x) = x + cos x. (3.150) u2 (x) = 1 + x −

Standard methods for solving ODEs The ODE (3.169) is of second order and nonhomogeneous. The auxiliary equation for the homogeneous part is of the form

3.2 Volterra Integral Equations of the Second Kind

91

r2 + 1 = 0,

(3.151)

2

that gives r = ±i, i = −1. The general solution is given by u(x) = uc + up , (3.152) u(x) = A cos x + B sin x + a + bx, where uc is the complementary solution, and up is a particular solution. Using ODE methods and initial conditions we ﬁnd that B = a = 0, and A = b = 1. The particular solution is given by u(x) = x + cos x.

(3.153)

Example 3.18 Solve the Volterra integral equation by using the variational iteration method 1 2 1 x u(x) = 1 + x + x + (x − t)2 u(t)dt. (3.154) 2 2 0 Using Leibnitz rule to diﬀerentiate both sides of (3.154) three times with respect to x gives the two integro-diﬀerential equations x (x − t)u(t)dt, u(0) = 1 u (x) = 1 + x + 0 (3.155) x u (x) = 1 + u(t)dt, u(0) = 1, u (0) = 1. 0

and the third order initial value problem u (x) = u(x), u(0) = u (0) = u (0) = 1. Using the variational iteration method VIM

(3.156)

(i) We ﬁrst note that we obtained two equivalent integro-diﬀerential equations (3.155). We will apply the VIM to these two equations. We ﬁrst start using the VIM to handle the integro-diﬀerential equation x (x − t)u(t)dt, u(0) = 1. (3.157) u (x) = 1 + x + 0

The correction functional for Eq. (3.157) is x ξ λ(ξ) un (ξ) − 1 − ξ − (ξ − r)˜ un (r)dr dξ. un+1 (x) = un (x) + 0

0

(3.158) Proceeding as before we ﬁnd λ = −1,

(3.159)

that gives the iteration formula x ξ un (ξ) − 1 − ξ − un+1 (x) = un (x) − (ξ − r)un (r)dr dξ. 0

(3.160)

0

We can use the initial conditions to select u0 (x) = u(0) = 1. Using this selection into (3.160) gives the following successive approximations

92

3 Volterra Integral Equations

u0 (x) = 1, u1 (x) = 1 −

x

0

= 1+x+ u2 (x) = 1 −

x

0

u0 (ξ)

−1−ξ−

1 1 2 x + x3 , 2! 3! u1 (ξ)

−1−ξ−

(ξ − r)u0 (r)dr dξ

0

0

ξ

ξ

(3.161)

(ξ − r)u1 (r)dr dξ

1 1 1 1 1 = 1 + x + x2 + x3 + x4 + x5 + x6 , 2! 3! 4! 5! 6! .. . 1 2 1 1 1 1 1 x + x3 + x4 + x5 + x6 + · · · + xn . 2! 3! 4! 5! 6! n! This in turn gives the exact solution by un (x) = 1 + x +

u(x) = ex .

(3.162)

(ii) We next consider the integro-diﬀerential equation x u(t)dt, u(0) = 1, u (0) = 1. u (x) = 1 +

(3.163)

0

The correction functional for Eq. (3.163) is x λ(ξ) un (ξ) − 1 − un+1 (x) = un (x) + 0

ξ 0

u ˜n (r)dr

dξ.

(3.164)

Notice that the integro-diﬀerential equation is of second order. Therefore, we can show that λ = ξ − x, (3.165) that gives the iteration formula x ξ (ξ − r)un (r)dr) dξ. (ξ − x)(un (ξ) − 1 − un+1 (x) = un (x) + 0

0

(3.166) We can use the initial conditions to select u0 (x) = 1 + x. Using this selection into (3.166) gives the following successive approximations u0 (x) = 1 + x, 1 2 1 1 x + x3 + x4 , 2! 3! 4! 1 1 1 u2 (x) = 1 + x + x2 + x3 + x4 + 2! 3! 4! .. . 1 1 1 un (x) = 1 + x + x2 + x3 + x4 + 2! 3! 4! This in turn gives the exact solution by u1 (x) = 1 + x +

1 5 1 1 x + x6 + x7 , 5! 6! 7! 1 5 1 1 x + x6 + · · · + xn . 5! 6! n!

(3.167)

3.2 Volterra Integral Equations of the Second Kind

u(x) = ex .

93

(3.168)

(iii) We next use the variational iteration method for solving the initial value problem u (x) − u(x) = 0, u(0) = u (0) = u (0) = 1. (3.169) x λ(ξ) (u ˜n (ξ)) dξ. (3.170) un+1 (x) = un (x) + n (ξ) − u 0

Using the formula (iii) given above for λ leads to 1 λ = − (ξ − x)2 . (3.171) 2! Substituting this value of the Lagrange multiplier into the functional (3.170) gives the iteration formula 1 x un+1 (x) = un (x) − (ξ − x)2 (u (3.172) n (ξ) − un (ξ)) dξ. 2! 0 As stated before, we can use the initial conditions to select 1 1 (3.173) u0 (x) = u(0) + xu (0) + u (0) = 1 + x + x2 . 2 2! Using this selection into (3.172) gives the following successive approximations 1 u0 (x) = 1 + x + x2 , 2! 1 1 1 1 u1 (x) = 1 + x + x2 + x3 + x4 + x5 , 2! 3! 4! 5! 1 2 1 3 1 4 1 1 1 1 u2 (x) = 1 + x + x + x + x + x5 + x6 + x7 + x8 , (3.174) 2! 3! 4! 5! 6! 7! 8! .. . 1 1 1 1 1 1 1 un (x) = 1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x8 + · · · 2! 3! 4! 5! 6! 7! 8! The VIM admits the use of u(x) = lim un (x), n→∞

(3.175)

that gives the exact solution by (3.176) u(x) = ex . Using standard methods for solving ODEs The ODE (3.169) is of third order, therefore the auxiliary equation is of the form r3 − 1 = 0, (3.177) that gives r = 1, − 21 ±

√

3 i, i2 2

= −1. The general solution is given by √ √ 3 3 x − 12 x x + C sin x . (3.178) B cos u(x) = Ae + e 2 2

To obtain the particular solution, we use the initial conditions to ﬁnd that the particular solution

94

3 Volterra Integral Equations

u(x) = ex .

(3.179)

It is interesting to point out that we need to use diﬀerent approaches to solve ODEs by standard methods, whereas the variational iteration method attacks all problems directly and in a straightforward manner.

Exercises 3.2.4 Use the variational iteration method to solve the following Volterra integral equations by converting the equation to initial value problem or to an equivalent integrodiﬀerential equation: x x 1 1 1. u(x) = 1 − u(t)dt 2. u(x) = x + x4 + x2 + x5 − u(t)dt 2 5 0 0 x x 1 1 u(t)dt 4. u(x) = 1 − x − x2 + (x − t)u(t)dt 3. u(x) = 1 − x2 + 2 2 0 0 x x 5. u(x) = 1 − (x − t)u(t)dt 6. u(x) = x + (x − t)u(t)dt 0

0

7. u(x) = 1 + 2x + 4

x

0

(x − t)u(t)dt 8. u(x) = 5 + 2x2 −

1 2 1 x x + (x − t)2 u(t)dt 2 2 0 1 x 1 10. u(x) = 1 + x + (x − t + 1)u(t)dt 2 2 0 x (x − t + 1)2 u(t)dt 11. u(x) = 1 + x2 −

x

0

(x − t)u(t)dt

9. u(x) = 1 + x +

0

1 1 12. u(x) = 1 − x2 + 2 6

x

0

(x − t)3 u(t)dt

13. u(x) = 2 + x − 2 cos x − 14. u(x) = 1 − 2 sinh x + 15. u(x) = 1 +

x

0

x 0

(x − t + 2)u(t)dt

(x − t + 2)u(t)dt

1 2 3 1 x+ x + 5 10 10

x

(x − t + 2)2 u(t)dt

0

3 2 1 3 1 x + x − 2 6 2 x 17. u(x) = 1 − x sin x + tu(t)dt

16. u(x) = 2 + 3x +

0

19. u(x) = −1 + ex +

1 2 x 1 x e − 2 2

20. u(x) = 1 − x sin x + x cos x +

0

(x − t + 2)2 u(t)dt

18. u(x) = x cosh x −

x

0 x 0

x

tu(t)dt tu(t)dt

x 0

tu(t)dt

3.2 Volterra Integral Equations of the Second Kind

95

3.2.5 The Successive Approximations Method The successive approximations method, also called the Picard iteration method provides a scheme that can be used for solving initial value problems or integral equations. This method solves any problem by ﬁnding successive approximations to the solution by starting with an initial guess, called the zeroth approximation. As will be seen, the zeroth approximation is any selective real-valued function that will be used in a recurrence relation to determine the other approximations. Given the linear Volterra integral equation of the second kind x u(x) = f (x) + λ K(x, t)u(t)dt, (3.180) 0

where u(x) is the unknown function to be determined, K(x, t) is the kernel, and λ is a parameter. The successive approximations method introduces the recurrence relation x

un (x) = f (x) + λ

0

K(x, t)un−1 (t)dt, n 1,

(3.181)

where the zeroth approximation u0 (x) can be any selective real valued function. We always start with an initial guess for u0 (x), mostly we select 0, 1, x for u0 (x), and by using (3.181), several successive approximations uk , k 1 will be determined as x

u1 (x) = f (x) + λ

u2 (x) = f (x) + λ

.. . un (x) = f (x) + λ

x

K(x, t)u1 (t)dt,

0

u3 (x) = f (x) + λ

K(x, t)u0 (t)dt,

0

x

K(x, t)u2 (t)dt,

0

0

(3.182)

x

K(x, t)un−1 (t)dt.

The question of convergence of un (x) is justiﬁed by noting the following theorem. Theorem 3.1 If f (x) in (3.181) is continuous for the interval 0 x a, and the kernel K(x, t) is also continuous in the triangle 0 x a, 0 t x, the sequence of successive approximations un (x), n 0 converges to the solution u(x) of the integral equation under discussion. It is interesting to point out that the variational iteration method admits the use of the iteration formula: x ∂un (ξ) −u ˜n (ξ) dξ. λ(ξ) (3.183) un+1 (x) = un (x) + ∂ξ 0 whereas the successive approximations method uses the iteration formula

96

3 Volterra Integral Equations

un (x) = f (x) + λ

x 0

K(x, t)un−1 (t)dt, n 1.

(3.184)

The diﬀerence between the two formulas can be summarized as follows: 1. The ﬁrst formula contains the Lagrange multiplier λ that should be determined ﬁrst before applying the formula. The successive approximations formula does not require the use of λ. 2. The ﬁrst variational iteration formula allows the use of the restriction u ˜n (ξ) where δ˜ un (ξ) = 0. The second formula does require this restriction. 3. The ﬁrst formula is applied to an equivalent ODE of the integral equation, whereas the second formula is applied directly to the iteration formula of the integral equation itself. The successive approximations method, or the Picard iteration method will be illustrated by the following examples. Example 3.19 Solve the Volterra integral equation by using the successive approximations method x u(x) = 1 − (x − t)u(t)dt. (3.185) 0

For the zeroth approximation u0 (x), we can select (3.186) u0 (x) = 1. The method of successive approximations admits the use of the iteration formula x (x − t)un (t)dt, n 0. (3.187) un+1 (x) = 1 − 0

Substituting (3.186) into (3.187) we obtain x 1 (x − t)u0 (t)dt = 1 − x2 , u1 (x) = 1 − 2! 0 x 1 u2 (x) = 1 − (x − t)u1 (t)dt = 1 − x2 + 2! 0 x 1 (x − t)u2 (t)dt = 1 − x2 + u3 (x) = 1 − 2! 0 x 1 (x − t)u3 (t)dt = 1 − x2 + u4 (x) = 1 − 2! 0 .. .

1 4 x , 4! 1 4 1 x − x6 , 4! 6!

(3.188)

1 4 1 1 x − x6 + x8 , 4! 6! 8!

Consequently, we obtain un+1 (x) =

n

(−1)k

k=0

x2k . (2k)!

(3.189)

The solution u(x) of (3.185) u(x) = lim un+1 (x) = cos x. n→∞

(3.190)

3.2 Volterra Integral Equations of the Second Kind

97

Example 3.20 Solve the Volterra integral equation by using the successive approximations method 1 1 x u(x) = 1 + x + x2 + (x − t)2 u(t)dt. (3.191) 2 2 0 For the zeroth approximation u0 (x), we can select u0 (x) = 0.

(3.192)

The method of successive approximations admits the use of the iteration formula 1 1 x (x − t)2 un(t)dt, n 0. (3.193) un+1 (x) = 1 + x + x2 + 2 2 0 Substituting (3.192) into (3.193) we obtain 1 u1 (x) = 1 + x + x2 , 2! 1 1 1 1 (3.194) u2 (x) = 1 + x + x2 + x3 + x4 + x5 , 2! 3! 4! 5! 1 1 1 1 1 1 1 u3 (x) = 1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x8 , 2! 3! 4! 5! 6! 7! 8! .. . and so on. The solution u(x) of (3.191) is given by (3.195) u(x) = lim un+1 (x) = ex . n→∞

Example 3.21 Solve the Volterra integral equation by using the successive approximations method 1 1 x u(x) = −1 + ex + x2 ex − tu(t)dt. (3.196) 2 2 0 For the zeroth approximation u0 (x), we select u0 (x) = 0. We next use the iteration formula 1 1 x tun (t)dt, n 0. un+1 (x) = −1 + ex + x2 ex − 2 2 0 Substituting (3.197) into (3.198) we obtain 1 u1 (x) = −1 + ex + x2 ex , 2! 1 5 1 u2 (x) = −3 + x2 + ex (3 − 2x + x2 − x3 ), 4 4 4

(3.197) (3.198)

98

3 Volterra Integral Equations

1 2 1 3 1 4 1 5 u3 (x) = x 1 + x + x + x + x + x + · · · , (3.199) 2! 3! 4! 5! .. . 1 2 1 3 1 4 1 5 1 7 un+1 (x) = x 1 + x + x + x + x + x + x + · · · . 2! 3! 4! 5! 7! x Notice that we used the Taylor expansion for e to determine u3 (x), u4 (x), . . . The solution u(x) of (3.196) u(x) = lim un+1 (x) = xex . n→∞

(3.200)

Example 3.22 Solve the Volterra integral equation by using the successive approximations method x tu(t)dt. (3.201) u(x) = 1 − x sin x + x cos x + 0

For the zeroth approximation u0 (x), we may select u0 (x) = x. We next use the iteration formula un+1 (x) = 1 − x sin x + x cos x +

0

(3.202) x

tun (t)dt, n 0.

(3.203)

Substituting (3.202) into (3.203) gives 1 u1 (x) = 1 + x3 − x sin x + x cos x, 3 1 1 u2 (x) = 3 + x2 + x3 − (2 + 3x − x2 ) sin x − (2 − 3x − x2 ) cos x, 2 15 1 2 1 3 1 5 1 7 1 4 1 6 u3 (x) = x − x + x − x + 1 − x + x − x , 3! 5! 7! 2! 4! 6! .. . n n x2k+1 x2k + . un+1 (x) = (−1)k (−1)k (2k + 1)! (2k)! k=0 k=0 (3.204) Notice that we used the Taylor expansion for sin x and cos x to determine the approximations u3 (x), u4 (x), . . .. The solution u(x) of (3.201) is given by u(x) = lim un+1 (x) = sin x + cos x. n→∞

(3.205)

Exercises 3.2.5 Use the successive approximations method to solve the following Volterra integral equations: x x 1. u(x) = x + u(t)dt 2. u(x) = x + (x − t)u(t)dt 0

0

3.2 Volterra Integral Equations of the Second Kind 99 x x 1 3. u(x) = x3 − (x − t)u(t)dt 4. u(x) = 1 + 2x + 4 (x − t)u(t)dt 6 0 0 x x 1 5. u(x) = x3 + (x − t)u(t)dt 6. u(x) = 1 + x2 − (x − t + 1)2 u(t)dt 6 0 0 x 1 1 1 x 7. u(x) = x2 − (x − t)u(t)dt 8. u(x) = 1 − x2 + (x − t)3 u(t)dt 2 2 6 0 0 x x u(t)dt 10. u(x) = 1 − 2 sinh x + (x − t + 2)u(t)dt 9. u(x) = 1 + 3 0

2

11. u(x) = 3 + x −

x

0

13. u(x) = x cosh x − 15. u(x) = 1 −

x

0

(x − t)u(t)dt 12. u(x) = 1 − x sin x +

x

tu(t)dt

0

3t2 u(t)dt

14. u(x) = 1 − x −

18. u(x) = 1 + sinh x + sin x − cos x + cosh x −

x

0

20. u(x) = −x + 2 sinh x +

x 0

tu(t)dt

0

(x − t)u(t)dt

16. u(x) = 2x cosh x − 4

17. u(x) = 1 + sinh x − sin x + cos x − cosh x +

19. u(x) = 2 − 2 cos x −

0 x

x

0

tu(t)dt

x

0 x 0

u(t)dt u(t)dt

(x − t)u(t)dt

x 0

(x − t)u(t)dt

3.2.6 The Laplace Transform Method The Laplace transform method is a powerful technique that can be used for solving initial value problems and integral equations as well. We assume that the reader has used the Laplace transform method, and the inverse Laplace transform, for solving ordinary diﬀerential equations. The details and properties of the Laplace method can be found in ordinary diﬀerential equations texts. Before we start applying this method, we summarize some of the concepts presented in Section 1.5. In the convolution theorem for the Laplace transform, it was stated that if the kernel K(x, t) of the integral equation: x u(x) = f (x) + λ K(x, t)u(t)dt, (3.206) 0

depends on the diﬀerence x − t, then it is called a diﬀerence kernel. Examples of the diﬀerence kernel are ex−t , cos(x − t), and x − t. The integral equation can thus be expressed as x K(x − t)u(t)dt. (3.207) u(x) = f (x) + λ 0

100

3 Volterra Integral Equations

Consider two functions f1 (x) and f2 (x) that possess the conditions needed for the existence of Laplace transform for each. Let the Laplace transforms for the functions f1 (x) and f2 (x) be given by L{f1 (x)} = F1 (s), (3.208) L{f2 (x)} = F2 (s). The Laplace convolution product of these two functions is deﬁned by x f1 (x − t)f2 (t)dt, (3.209) (f1 ∗ f2 )(x) = 0

or (f2 ∗ f1 )(x) =

0

x

f2 (x − t)f1 (t)dt.

(3.210)

Recall that (f1 ∗ f2 )(x) = (f2 ∗ f1 )(x).

(3.211)

We can easily show that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x L{(f1 ∗ f2 )(x)} = L f1 (x − t)f2 (t)dt = F1 (s)F2 (s). (3.212) 0

Based on this summary, we will examine speciﬁc Volterra integral equations where the kernel is a diﬀerence kernel. Recall that we will apply the Laplace transform method and the inverse of the Laplace transform using Table 1.1 in Section 1.5. By taking Laplace transform of both sides of (3.207) we ﬁnd U (s) = F (s) + λK(s)U (s),

(3.213)

where U (s) = L{u(x)}, K(s) = L{K(x)}, F (s) = L{f (x)}. (3.214) Solving (3.213) for U (s) gives F (s) , λK(s) = 1. (3.215) U (s) = 1 − λK(s) The solution u(x) is obtained by taking the inverse Laplace transform of both sides of (3.215) where we ﬁnd F (s) −1 u(x) = L . (3.216) 1 − λK(s) Recall that the right side of (3.216) can be evaluated by using Table 1.1 in Section 1.5. The Laplace transform method for solving Volterra integral equations will be illustrated by studying the following examples. Example 3.23 Solve the Volterra integral equation by using the Laplace transform method x u(t)dt. (3.217) u(x) = 1 + 0

3.2 Volterra Integral Equations of the Second Kind

101

Notice that the kernel K(x − t) = 1, λ = 1. Taking Laplace transform of both sides (3.217) gives L{u(x)} = L{1} + L{1 ∗ u(x)}, (3.218) so that U (s) =

1 1 + U (s), s s

(3.219)

1 . s−1

(3.220)

or equivalently U (s) =

By taking the inverse Laplace transform of both sides of (3.220), the exact solution is therefore given by (3.221) u(x) = ex . Example 3.24 Solve the Volterra integral equation by using the Laplace transform method x (x − t)u(t)dt. (3.222) u(x) = 1 − 0

Notice that the kernel K(x − t) = (x − t), λ = −1. Taking Laplace transform of both sides (3.222) gives L{u(x)} = L{1} − L{(x − t) ∗ u(x)}, so that U (s) =

(3.223)

1 1 − 2 U (s), s s

(3.224)

s . s2 + 1

(3.225)

or equivalently U (s) =

By taking the inverse Laplace transform of both sides of (3.225), the exact solution u(x) = cos x, (3.226) is readily obtained. Example 3.25 Solve the Volterra integral equation by using the Laplace transform x 1 (x − t)u(t)dt. u(x) = x3 − 3! 0 Taking Laplace transform of both sides (3.227) gives 1 L{u(x)} = L{x3 } − L{(x − t) ∗ u(x)}. 3! This gives 3! 1 1 × 4 − 2 U (s), U (s) = 3! s s so that

method (3.227)

(3.228)

(3.229)

102

3 Volterra Integral Equations

U (s) =

1 1 1 = 2 − 2 . s2 (s2 + 1) s s +1

(3.230)

Taking the inverse Laplace transform of both sides of (3.230) gives the exact solution u(x) = x − sin x. (3.231) Example 3.26 Solve the Volterra integral equation by using the Laplace transform method x sin(x − t)u(t)dt. (3.232) u(x) = sin x + cos x + 2 0

Recall that we should use the linear property of the Laplace transforms. Taking Laplace transform of both sides (3.232) gives L{u(x)} = L{sin x + cos x} + 2L{sin(x − t) ∗ u(x)}, (3.233) so that U (s) =

s2

s 2 1 + 2 + 2 U (s), +1 s +1 s +1

(3.234)

or equivalently U (s) =

1 . s−1

(3.235)

Taking the inverse Laplace transform of both sides of (3.235) gives the exact solution (3.236) u(x) = ex .

Exercises 3.2.6 Use the Laplace transform method to solve the Volterra integral equations: x x 1. u(x) = x + (x − t)u(t)dt 2. u(x) = 1 − x − (x − t)u(t)dt 0

1 x 1 (x − t)3 u(t)dt 3. u(x) = 1 − x2 + 2 6 0 x 5. u(x) = x − 1 + (x − t)u(t)dt 0

6. u(x) = cos x − sin x + 2 7. u(x) = ex − cos x − 2

x

0 x

0

0

4. u(x) = 1 + 3

x 0

(x − t)u(t)dt

cos(x − t)u(t)dt

ex−t u(t)dt

9. u(x) = sin x + sinh x + cosh x − 2

x

cos (x − t)u(t)dt

0

10. u(x) = sinh x + cosh x − cos x − 2 11. u(x) = sin x − cos x + cosh x − 2

8. u(x) = 1 −

x

0 x

0

cos (x − t)u(t)dt

cosh (x − t)u(t)dt

x

0

(x − t)2 − 1 u(t)dt

3.2 Volterra Integral Equations of the Second Kind x 12. u(x) = sin x + cos x + sinh x − 2 cosh (x − t)u(t)dt

13. u(x) = 2ex − 2 − x + 15. u(x) = 2 − 2 cos x −

0

x

0 x

0

103

(x − t)u(t)dt 14. u(x) = 2 cosh x − 2 +

(x − t)u(t)dt

16. u(x) = 1 +

x 0

x

0

(x − t)u(t)dt

sin(x − t)u(t)dt

3.2.7 The Series Solution Method A real function u(x) is called analytic if it has derivatives of all orders such that the Taylor series at any point b in its domain n f (k) (b) (x − b)k , (3.237) u(x) = k! k=0

converges to f (x) in a neighborhood of b. For simplicity, the generic form of Taylor series at x = 0 can be written as ∞ an xn . (3.238) u(x) = n=0

In this section we will present a useful method, that stems mainly from the Taylor series for analytic functions, for solving Volterra integral equations. We will assume that the solution u(x) of the Volterra integral equation x K(x, t)u(t)dt, (3.239) u(x) = f (x) + λ 0

is analytic, and therefore possesses a Taylor series of the form given in (3.238), where the coeﬃcients an will be determined recurrently. Substituting (3.238) into both sides of (3.239) gives ∞ x ∞ n n an x = T (f (x)) + λ K(x, t) an t dt, (3.240) 0

n=0

or for simplicity we use 2

a0 + a1 x + a2 x + · · · = T (f (x)) + λ

n=0

0

x

K(x, t) a0 + a1 t + a2 t2 + · · · dt,

(3.241) where T (f (x)) is the Taylor series for f (x). The integral equation (3.239) will be converted to a traditional integral in (3.240) or (3.241) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integral in (3.240) or (3.241), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of

104

3 Volterra Integral Equations

like powers of x in both sides of the resulting equation to obtain a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (3.238). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. Example 3.27 Solve the Volterra integral equation by using the series solution method x u(x) = 1 + u(t)dt. (3.242) 0

Substituting u(x) by the series u(x) =

∞

an xn ,

(3.243)

n=0

into both sides of Eq. (3.242) leads to ∞ an xn = 1 + n=0

x

0

∞

an t

n

dt.

Evaluating the integral at the right side gives ∞ ∞ 1 an xn+1 , an xn = 1 + n + 1 n=0 n=0 that can be rewritten as a0 +

∞ n=1

a n xn = 1 +

(3.244)

n=0

∞ 1 an−1 xn , n n=1

(3.245)

(3.246)

or equivalently 1 1 a0 + a1 x + a2 x2 + a3 x3 + · · · = 1 + a0 x + a1 x2 + a2 x2 + · · · . (3.247) 2 3 In (3.245), the powers of x of both sides are diﬀerent, therefore, we make them the same by changing the index of the second sum to obtain (3.246). Equating the coeﬃcients of like powers of x in both sides of (3.246) gives the recurrence relation a0 = 1, (3.248) 1 an = an−1 , n 1. n where this result gives 1 (3.249) an = , n 0. n! Substituting this result into (3.243) gives the series solution:

3.2 Volterra Integral Equations of the Second Kind

105

∞ 1 n x , n! n=0

(3.250)

u(x) =

that converges to the exact solution u(x) = ex . It is interesting to point out that this result can be obtained by equating coeﬃcients of like terms in both sides of (3.247), where we ﬁnd a0 = 1, a1 = a0 = 1, 1 1 a 2 = a1 = , 2 2! (3.251) .. . 1 1 an−1 = . n n! This leads to the same result obtained before by solving the recurrence relation. an =

Example 3.28 Solve the Volterra integral equation by using the series solution method x (x − t)u(t)dt. (3.252) u(x) = x + 0

Substituting u(x) by the series u(x) =

∞

an xn ,

(3.253)

n=0

into both sides of Eq. (3.252) leads to x ∞ ∞ ∞ an xn = x + xan tn − an tn+1 dt. 0

n=0

n=0

Evaluating the right side leads to ∞ ∞ an xn = x +

1 an xn+2 , (n + 2)(n + 1) n=0

n=0

that can be rewritten as a0 + a1 x +

∞ n=2

an xn = x +

(3.254)

n=0

∞

1 an−2 xn , n(n − 1) n=2

(3.255)

(3.256)

or equivalently 1 1 1 a0 + a1 x + a2 x2 + a3 x3 + · · · = x + a0 x2 + a1 x3 + a2 x4 + · · · (3.257) 2 6 12 Equating the coeﬃcients of like powers of x in both sides of (3.256) gives the recurrence relation a0 = 0, a1 = 1, .. . (3.258) 1 an−2 , n 2. an = n(n − 1)

106

3 Volterra Integral Equations

This result can be combined to obtain 1 , n 0. an = (2n + 1)! Substituting this result into (3.253) gives the series solution ∞ 1 u(x) = x2n+1 , (2n + 1)! n=0 that converges to the exact solution u(x) = sinh x.

(3.259)

(3.260)

(3.261)

It is interesting to point out that this result can also be obtained by equating coeﬃcients of like terms in both sides of (3.257), where we ﬁnd 1 a0 = 0, a1 = 1, a2 = a0 = 0, 2 (3.262) 1 1 1 a2 = 0. a3 = a1 = , a4 = 6 3! 12 This leads to the same result obtained before by solving the recurrence relation. Example 3.29 Solve the Volterra integral equation by using the series solution method x tu(t)dt. (3.263) u(x) = 1 − x sin x + 0

For simplicity reasons, we will use few terms of the Taylor series for sin x and for the solution u(x) in (3.263) to ﬁnd a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · x (3.264) x3 =1−x x− t(a0 + a1 t + a2 t2 + · · · )dt. + ··· + 3! 0 Integrating the right side and collecting the like terms of x we ﬁnd 1 1 1 1 a0 +a1 x+a2 x2 +a3 x3 +a4 x4 +· · · = 1+( a0 −1)x2 + a1 x3 +( + a2 )x4 +· · · . 2 3 6 4 (3.265) Equating the coeﬃcients of like powers of x in (3.265) yields a1 = 0, a0 = 1, 1 1 1 a2 = a0 − 1 = − , a3 = a1 = 0, 2 2! 3 (3.266) 1 1 1 a 4 = + a2 = , 6 4 4! .. . and generally a2n+1 = 0,

a2n =

(−1)n , (2n)!

n 0.

(3.267)

3.2 Volterra Integral Equations of the Second Kind

107

The solution in a series form is given by 1 1 1 u(x) = 1 − x2 + x4 − x6 + · · · 2! 4! 6! that gives the exact solution by

(3.268)

u(x) = cos x.

(3.269)

Example 3.30 Solve the Volterra integral equation by using the series solution method x u(x) = 2ex − 2 − x + (x − t)u(t)dt. (3.270) 0

Proceeding as before, we will use few terms of the Taylor series for ex and for the solution u(x) in (3.270) to ﬁnd a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · x 1 1 (x − t)(a0 + a1 t + a2 t2 + · · · )dt. = x + x2 + x3 + x4 + · · · + 3 12 0 (3.271) Integrating the right side and collecting the like terms of x we ﬁnd a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · 1 1 1 1 1 + a1 x3 + + a2 x4 + · · · . = x + 1 + a 0 x2 + 2 3 6 12 12 (3.272) Equating the coeﬃcients of like powers of x in (3.272) yields 1 1 a0 = 0, a1 = 1, a2 = 1, a3 = , a4 = , . . . (3.273) 2! 3! and generally 1 an = , n 1, a0 = 0. (3.274) n! The solution in a series form is given by 1 1 1 (3.275) u(x) = x(1 + x + x2 + x3 + x4 + · · · ), 2! 3! 4! that converges to the exact solution u(x) = xex .

(3.276)

Exercises 3.2.7 Use the series solution method to solve the Volterra integral equations: x x 1. u(x) = 1 − u(t)dt 2. u(x) = 1 − (x − t)u(t)dt 0

3. u(x) = x +

0

x

0

(x − t)u(t)dt

5. u(x) = 1 + xex −

x

0

tu(t)dt

1 1 4. u(x) = 1 + x + 2 2 6. u(x) = 1 + 2x + 4

x

0 x 0

(x − t + 1)u(t)dt

(x − t)u(t)dt

108 7. u(x) = 3 + x2 −

x

0 x

9. u(x) = x cos x +

0

(x − t)u(t)dt

12. u(x) = 1 − x −

x 0

10. u(x) = x cosh x −

tu(t)dt

11. u(x) = 2 cosh x − 2 +

3 Volterra Integral Equations x 8. u(x) = 1 + 2 sin x − u(t)dt

x

0

0 x

0

tu(t)dt

(x − t)u(t)dt

(x − t)u(t)dt

13. u(x) = x − x ln(1 + x) +

x 1 3 2xu(t)dt x + x3 ln(1 + x) − 2 0 x 15. u(x) = sec x + tan x − sec tu(t)dt 16. u(x) = x +

14. u(x) = x2 −

0

x

0

u(t)dt

x

0

tan tu(t)dt

3.3 Volterra Integral Equations of the First Kind The standard form of the Volterra integral equations of the ﬁrst kind is given by x

K(x, t)u(t)dt,

f (x) =

(3.277)

0

where the kernel K(x, t) and the function f (x) are given real-valued functions, and u(x) is the function to be determined. Recall that the unknown function u(x) appears inside and outside the integral sign for the Volterra integral equations of the second kind, whereas it occurs only inside the integral sign for the Volterra integral equations of the ﬁrst kind. This equation of the ﬁrst kind motivated mathematicians to develop reliable methods for solving it. In this section we will discuss three main methods that are commonly used for handling the Volterra integral equations of the ﬁrst kind. Other methods are available in the literature but will not be presented in this text.

3.3.1 The Series Solution Method As in the previous section, we will consider the solution u(x) to be analytic, where it has derivatives of all orders, and it possesses Taylor series at x = 0 of the form ∞ u(x) = a n xn , (3.278) n=0

where the coeﬃcients an will be determined recurrently. Substituting (3.278) into (3.277) gives

3.3 Volterra Integral Equations of the First Kind

T (f (x)) =

x

K(x, t) 0

∞

109

an tn dt,

(3.279)

n=0

or for simplicity we can use x

K(x, t) a0 + a1 t + a2 t2 + · · · dt, T (f (x)) =

(3.280)

0

where T (f (x)) is the Taylor series for f (x). The integral equation (3.277) will be converted to a traditional integral in (3.279) or (3.280) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. The method is identical to that presented before for the Volterra integral equations of the second kind. We ﬁrst integrate the right side of the integral in (3.279) or (3.280), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x in both sides of the resulting equation to obtain a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (3.278). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. This method will be illustrated by discussing the following examples. Example 3.31 Solve the Volterra integral equation by using the series solution method x tu(t)dt. (3.281) sin x − x cos x = 0

Proceeding as before, only few terms of the Taylor series for sin x − x cos x and for the solution u(x) in (3.281) will be used. Integrating the right side we obtain x 1 3 1 5 1 7 x − x + x + ··· = t(a0 + a1 t + a2 t2 + a3 t3 + · · · )dt, 3 30 840 0 1 1 1 1 = a0 x2 + a1 x3 + a2 x4 + a3 x5 2 3 4 5 1 1 + a4 x6 + a5 x7 + · · · 6 7 (3.282) Equating the coeﬃcients of like powers of x in (3.282) yields 1 1 (3.283) a0 = 0, a1 = 1, a2 = 0, a3 = − , a4 = 0, a5 = · · · 3! 5! The solution in a series form is given by

110

3 Volterra Integral Equations

1 3 1 x + x5 + · · · 3! 5! that converges to the exact solution u(x) = sin x. u(x) = x −

(3.284) (3.285)

Example 3.32 Solve the Volterra integral equation by using the series solution method x (x − t)u(t)dt. (3.286) 2 + x − 2ex + xex = 0

Using few terms of the Taylor series for 2 + x − 2ex + xex and for the solution u(x) in (3.286), and by integrating the right side we ﬁnd x 1 1 1 3 x + x4 + x5 + · · · = (x − t)(a0 + a1 t + a2 t2 + · · · )dt 6 12 40 0 1 1 1 1 = a0 x2 + a1 x3 + a2 x4 + a3 x5 (3.287) 2 6 12 20 1 6 + a4 x + · · · 30 Equating the coeﬃcients of like powers of x in (3.287) yields 1 1 a0 = 0, a1 = 1, a2 = 1, a3 = , a4 = , · · · (3.288) 2! 3! The solution in a series form is given by 1 1 u(x) = x(1 + x + x2 + x3 + · · · ), (3.289) 2! 3! that converges to the exact solution (3.290) u(x) = xex . Example 3.33 Solve the Volterra integral equation by using the series solution method x 1 2 2 2tu(t)dt. (3.291) x − x − ln(1 + x) + x ln(1 + x) = 2 0 Using the Taylor series for x − 12 x2 − ln(1 + x) + x2 ln(1 + x) and proceeding as before we ﬁnd 2 1 2 3 1 4 x − x + x5 − x6 + · · · 3 4 15 12 x (3.292) = 2t(a0 + a1 t + a2 t2 + a3 t3 + · · · )dt 0

2 1 2 1 = a0 x2 + a1 x3 + a2 x4 + a3 x5 + a4 x6 + · · · 3 2 5 3 Equating the coeﬃcients of like powers of x in (3.292) yields 1 1 1 a0 = 0, a1 = 1, a2 = − , a3 = , a4 = − , . . . 2 3 4 The solution in a series form is given by

(3.293)

3.3 Volterra Integral Equations of the First Kind

111

1 1 1 u(x) = x − x2 + x3 − x4 + · · · 2 3 4 that converges to the exact solution u(x) = ln(1 + x).

(3.294) (3.295)

Exercises 3.3.1 Use the series solution method to solve the Volterra integral equations of the ﬁrst kind: x x (x − t + 1)u(t)dt 2. x cosh x − sinh x = u(t)dt 1. ex − 1 − x = 0

3. 1 + xex − ex = 5. − 1 − x +

0

x 0

1 4. 1 + x3 + xex − ex = 3

tu(t)dt

1 3 x + ex = 6

x

0

8. x − 2 sinh x + x cosh x = 9. 1 + x − sin x − cos x =

x 0 x

0 x

0

x

0

tu(t)dt

(x − t)u(t)dt

6. − x + 2 sin x − x cos x =

(x − t)u(t)dt

7. − 1 + cosh x =

x 0

(x − t)u(t)dt

(x − t)u(t)dt 10. 1 − x − e−x =

(x − t)u(t)dt

1 2 x + ln(1 + x) + x ln(1 + x) = 2 x 1 12. x2 ex = ex−t u(t)dt 2 0

11. − x +

x

0

(x − t)u(t)dt

x 0

u(t)dt

3.3.2 The Laplace Transform Method The Laplace transform method is a powerful technique that we used before for solving initial value problems and Volterra integral equations of the second kind. In the convolution theorem for the Laplace transform method, it was stated that if the kernel K(x, t) of the integral equation x K(x, t)u(t)dt, (3.296) f (x) = 0

depends on the diﬀerence x − t, then it is called a diﬀerence kernel. The Volterra integral equation of the ﬁrst kind can thus be expressed as x f (x) = K(x − t)u(t)dt. (3.297) 0

Consider two functions f1 (x) and f2 (x) that possess the conditions needed for the existence of Laplace transform for each. Let the Laplace transforms for the functions f1 (x) and f2 (x) be given by

112

3 Volterra Integral Equations

L{f1 (x)} = F1 (s), (3.298) L{f2 (x)} = F2 (s). The Laplace convolution product of these two functions is deﬁned by x (f1 ∗ f2 )(x) = f1 (x − t)f2 (t)dt, (3.299) 0

or (f2 ∗ f1 )(x) =

x

0

f2 (x − t)f1 (t)dt.

(3.300)

Recall that (f1 ∗ f2 )(x) = (f2 ∗ f1 )(x).

(3.301)

We can easily show that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x f1 (x − t)f2 (t)dt = F1 (s)F2 (s). (3.302) L{(f1 ∗ f2 )(x)} = L 0

Based on this summary, we will examine speciﬁc Volterra integral equations of the ﬁrst kind where the kernel is a diﬀerence kernel. Recall that we will apply the Laplace transform method and the inverse of the Laplace transform using Table 1.1 in Section 1.5. By taking Laplace transform of both sides of (3.297) we ﬁnd F (s) = K(s)U (s), (3.303) where U (s) = L{u(x)}, Solving (3.303) gives

K(s) = L{K(x)}, U (s) =

F (s) = L{f (x)}.

F (s) , K(s)

(3.304) (3.305)

where K(s) = 0.

(3.306)

The solution u(x) is obtained by taking the inverse Laplace transform of both sides of (3.305) where we ﬁnd F (s) . (3.307) u(x) = L−1 K(s) Recall that the right side of (3.307) can be evaluated by using Table 1.1 in Section 1.5. The Laplace transform method for solving Volterra integral equations of the ﬁrst kind will be illustrated by studying the following examples. Example 3.34 Solve the Volterra integral equation of the ﬁrst kind by using the Laplace transform method x x e − sin x − cos x = 2ex−tu(t)dt. (3.308) 0

Taking the Laplace transform of both sides of (3.308) yields

3.3 Volterra Integral Equations of the First Kind

113

1 s 2 1 − − = U (s), s − 1 s2 + 1 s2 + 1 s−1

(3.309)

2 2 = U (s). (s − 1)(s2 + 1) s−1

(3.310)

or equivalently

This in turn gives 1 . (3.311) +1 Taking the inverse Laplace transform of both sides gives the exact solution U (s) =

s2

u(x) = sin x.

(3.312)

Example 3.35 Solve the Volterra integral equation of the ﬁrst kind by using the Laplace transform method x x 1+x−e = (t − x)u(t)dt. (3.313) 0

Notice that the kernel is (t − x) = −(x − t). Taking the Laplace transform of both sides of (3.313) yields 1 1 1 1 + = − 2 U (s), − s s2 s−1 s so that 1 s(s − 1) + s − 1 − s2 = − 2 U (s). 2 s (s − 1) s

(3.314)

(3.315)

Solving for U (s) we ﬁnd 1 . (3.316) s−1 Taking the inverse Laplace transform of both sides gives the exact solution U (s) =

u(x) = ex .

(3.317)

Example 3.36 Solve the Volterra integral equation of the ﬁrst kind by using the Laplace transform method x 1 −1 + x2 + x3 + 2 sinh x + cosh x = (x − t + 2)u(t)dt. (3.318) 6 0 Taking the Laplace transform of both sides of (3.318) yields s3 + s2 − 1 1 2 = U (s), (3.319) + s2 (s2 − 1) s2 s or equivalently 1 s . (3.320) U (s) = 2 + 2 s s −1 Taking the inverse Laplace transform of both sides gives the exact solution u(x) = x + cosh x. (3.321)

114

3 Volterra Integral Equations

Exercises 3.3.2 Use the Laplace transform method to solve the Volterra integral equations of the ﬁrst kind: x x 1. x − sin x = (x − t)u(t)dt 2. ex + sin x − cos x = 2ex−t u(t)dt 0

0

x 1 3. 1 + x3 − cos x = (x − t)u(t)dt 3! 0 x (1 + 2(x − t))u(t)dt 5. x = 0 x

7. x =

0

9. 1 + x −

6. sinh x =

1 3 x − ex = 3!

x

0

x

0

x

0

(x − t)u(t)dt

ex−t u(t)dt

8. 1 − x − e−x =

(x − t + 1)u(t)dt

4. 1 + x − sin x − cos x =

x

0

(t − x)u(t)dt

(t − x)u(t)dt

x 1 2 1 x + x3 − sin x − cos x = (x − t + 1)u(t)dt 2 3! 0 x 11. 3 − 7x + x2 + sinh x − 3 cosh x = (x − t − 3)u(t)dt 10. 1 + x +

12. 1 − cos x =

0

x

0

cos(x − t)u(t)dt

3.3.3 Conversion to a Volterra Equation of the Second Kind In this section we will present a method that will convert Volterra integral equations of the ﬁrst kind to Volterra integral equations of the second kind. The conversion technique works eﬀectively only if K(x, x) = 0. Diﬀerentiating both sides of the Volterra integral equation of the ﬁrst kind x K(x, t)u(t)dt, (3.322) f (x) = 0

with respect to x, and using Leibnitz rule, we ﬁnd x Kx (x, t)u(t)dt. f (x) = K(x, x)u(x) + 0

(3.323)

Solving for u(x), provided that K(x, x) = 0, we obtain the Volterra integral equation of the second kind given by x f (x) 1 u(x) = (3.324) − Kx (x, t)u(t)dt. K(x, x) K(x, x) 0 Notice that the non-homogeneous term and the kernel have changed to f (x) g(x) = , (3.325) K(x, x) and

3.3 Volterra Integral Equations of the First Kind

G(x, t) = −

115

1 Kx (x, t), K(x, x)

(3.326)

respectively. Having converted the Volterra integral equation of the ﬁrst kind to the Volterra integral equation of the second kind, we then can use any method that was presented before. Because we solved the Volterra integral equations of the second kind by many methods, we will select distinct methods for solving the Volterra integral equation of the ﬁrst kind after reducing it to a second kind Volterra integral equation. Example 3.37 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x ex−t u(t)dt. (3.327) sinh x = 0

Diﬀerentiating both sides of (3.327) and using Leibnitz rule we obtain x cosh x = u(x) + ex−tu(t)dt, (3.328) 0

that gives the Volterra integral equation of the second kind x ex−tu(t)dt. u(x) = cosh x −

(3.329)

0

We select the Laplace transform method for solving this problem. Taking Laplace transform of both sides gives s 1 U (s) = 2 − U (s), (3.330) s −1 s−1 that leads to 1 . (3.331) U (s) = s+1 Taking the inverse Laplace transform of both sides gives the exact solution u(x) = e−x . (3.332) Example 3.38 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x (x − t + 1)u(t)dt. (3.333) 1 + sin x − cos x = 0

Diﬀerentiating both sides of (3.333) and using Leibnitz rule we obtain the Volterra integral equation of the second kind x u(t)dt. (3.334) u(x) = cos x + sin x − 0

We select the modiﬁed decomposition method for solving this problem. Therefore we set the modiﬁed recurrence relation

116

3 Volterra Integral Equations

u0 (x) = cos x,

u1 (x) = sin x − uk+1 (x) = −

0

0

x

u0 (t)dt = 0,

(3.335)

x

uk (t)dt = 0, k 1.

This gives the exact solution by u(x) = cos x.

(3.336)

Example 3.39 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x 9x2 + 5x3 = (10x − 10t + 6)u(t)dt. (3.337) 0

Diﬀerentiating both sides of (3.337) and using Leibnitz rule we obtain x 10u(t)dt, (3.338) 18x + 15x2 = 6u(x) + 0

that gives the Volterra integral equation of the second kind 5 5 x u(x) = 3x + x2 − u(t)dt. (3.339) 2 3 0 We select the Adomian decomposition method combined with the noise terms phenomenon for solving this problem. Therefore we set the recurrence relation 5 5 x uk (t)dt, k 0, (3.340) u0 (x) = 3x + x2 , uk+1 (x) = − 2 3 0 that gives 5 5 25 u0 (x) = 3x + x2 , u1 (x) = − x2 − x3 . (3.341) 2 2 18 Canceling the noise term 52 x2 , that appear in u0 (x) and u1 (x), from u0 (x) gives the exact solution by u(x) = 3x. (3.342) Remarks 1. It was stated before that if K(x, x) = 0, then the conversion of the ﬁrst kind to the second kind fails. However, if K(x, x) = 0 and Kx (x, x) = 0, then by diﬀerentiating the Volterra integral equation of the ﬁrst kind as many times as needed, provided that K(x, t) is diﬀerentiable, then the equation will be reduced to the Volterra integral equation of the second kind. 2. The function f (x) must satisfy speciﬁc conditions to guarantee a unique continuous solution for u(x). The determination of these special conditions will be left as an exercise.

3.3 Volterra Integral Equations of the First Kind

117

However, for the ﬁrst remark, where K(x, x) = 0 but Kx (x, x) = 0, we will diﬀerentiate twice, by using Leibnitz rule, as will be shown by the following illustrative example. Example 3.40 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x x sinh x = 2 sinh(x − t)u(t)dt. (3.343) 0

Diﬀerentiating both sides of (3.343) and using Leibnitz rule we obtain x x cosh x + sinh x = 2 cosh(x − t)u(t)dt, (3.344) 0

which is still a Volterra integral equation of the ﬁrst kind. However, because Kx (x, x) = 0, we diﬀerentiate again to obtain the Volterra integral equation of the second kind x 1 sinh(x − t)u(t)dt. (3.345) u(x) = cosh x + x sinh x − 2 0 We select the modiﬁed decomposition method for solving this problem. Therefore we set the modiﬁed recurrence relation u0 (x) = cosh x, x 1 sinh(x − t)u0 (t)dt = 0, u1 (x) = x sinh x − 2 (3.346) 0 x uk+1 (x) = − sinh(x − t)uk (t)dt = 0, k 1. 0

The exact solution is given by u(x) = cosh x.

(3.347)

Exercises 3.3.3 In Exercises 1–12, use Leibnitz rule to convert the Volterra integral equation of the ﬁrst kind to a second kind and solve the resulting equation: x x 1. ex + sin x − cos x = 2ex−t u(t)dt 2. ex − cos x = ex−t u(t)dt 3. x =

0

x

0

(x − t + 1)u(t)dt

5. ex − x − 1 = 7. ex − 1 =

x 0

9. 5x4 + x5 =

x 0

(x − t + 1)u(t)dt

(x − t + 1)u(t)dt

x 0

(x − t + 1)u(t)dt

0

4. ex + sin x − cos x = 6.

1 2 x x e = 2

x

0

x 0

2 cos(x − t)u(t)dt

ex−t u(t)dt

8. sin x − cos x + 1 =

x 0

(x − t + 1)u(t)dt

118

3 Volterra Integral Equations

10. 4 + x − 4ex + 3xex =

x

0

(x − t + 2)u(t)dt

x 1 3 x + 3ex = (x − t + 2)u(t)dt 3! 0 x π 12. tan x − ln cos x = (x − t + 1)u(t)dt, x < 2 0 11. − 3 − x + x2 +

In Exercises 13–16, use Leibnitz rule twice to convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation: x x 13. x sin x = 2 sin(x − t)u(t)dt 14. ex − sin x − cos x = 2 sin(x − t)u(t)dt 0

15. sin x − cos x + e

−x

16. sin x − x cos x =

0

x

= 0 x

0

2 sin(x − t)u(t)dt

2 sinh(x − t)u(t)dt

References 1. M. Bocher, Integral Equations, Cambridge University Press, London, (1974). 2. P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, (1985). 3. B.L. Moiseiwitsch, Integral Equations, Longman, London and New York, (1977). 4. D. Porter and D.S. Stirling, Integral Equations: A Practical Treatment from Spectral Theory to Applications, Cambridge, (2004). 5. G. Adomian, Solving Frontier Problems of Physics: The decomposition method, Kluwer, (1994). 6. G. Adomian and R. Rach, Noise terms in decomposition series solution, Comput. Math. Appl., 24 (1992) 61–64. 7. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 8. A.M.Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 9. A.M. Wazwaz, A reliable modiﬁcation of the Adomian decomposition method, Appl. Math. Comput., 102 (1999) 77–86. 10. A.M. Wazwaz, Necessary conditions for the appearance of noise terms in decomposition solution series, Appl. Math. Comput., 81 (1997) 265–274. 11. J.H. He, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, 20 (2006) 1141–1199. 12. J.H. He, Variational iteration method for autonomous ordinary diﬀerential systems, Appl. Math. Comput., 114(2/3) (2000) 115–123.

Chapter 4

Fredholm Integral Equations

4.1 Introduction It was stated in Chapter 2 that Fredholm integral equations arise in many scientiﬁc applications. It was also shown that Fredholm integral equations can be derived from boundary value problems. Erik Ivar Fredholm (1866– 1927) is best remembered for his work on integral equations and spectral theory. Fredholm was a Swedish mathematician who established the theory of integral equations and his 1903 paper in Acta Mathematica played a major role in the establishment of operator theory. As stated before, in Fredholm integral equations, the integral containing the unknown function u(x) is characterized by ﬁxed limits of integration in the form b

K(x, t)u(t)dt,

u(x) = f (x) + λ

(4.1)

a

where a and b are constants. For the ﬁrst kind Fredholm integral equations, the unknown function u(x) occurs only under the integral sign in the form b f (x) = K(x, t)u(t)dt. (4.2) a

However, Fredholm integral equations of the second kind, the unknown function u(x) occurs inside and outside the integral sign. The second kind is represented by the form b u(x) = f (x) + λ K(x, t)u(t)dt. (4.3) a

The kernel K(x, t) and the function f (x) are given real-valued functions [9], and λ is a parameter. When f (x) = 0, the equation is said to be homogeneous. In this chapter, we will mostly use degenerate or separable kernels. A degenerate or a separable kernel is a function that can be expressed as the sum of the product of two functions each depends only on one variable. Such a kernel can be expressed in the form A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

120

4 Fredholm Integral Equations

K(x, t) =

n

fi (x)gi (t).

(4.4)

i=1

Examples of separable kernels are x − t, (x − t)2 , 4xt, etc. In what follows we state, without proof, the Fredholm alternative theorem. Theorem 4.1 (Fredholm Alternative Theorem) If the homogeneous Fredholm integral equation b K(x, t)u(t)dt (4.5) u(x) = λ a

has only the trivial solution u(x) = 0, then the corresponding nonhomogeneous Fredholm equation b u(x) = f (x) + λ K(x, t)u(t)dt (4.6) a

has always a unique solution. This theorem is known by the Fredholm alternative theorem [1]. Theorem 4.2 (Unique Solution) If the kernel K(x, t) in Fredholm integral equation (4.1) is continuous, real valued function, bounded in the square a x b and a t b, and if f (x) is a continuous real valued function, then a necessary condition for the existence of a unique solution for Fredholm integral equation (4.1) is given by |λ|M (b − a) < 1,

(4.7)

|K(x, t)| M ∈ R.

(4.8)

where

On the contrary, if the necessary condition (4.7) does not hold, then a continuous solution may exist for Fredholm integral equation. To illustrate this, we consider the Fredholm integral equation 1 (3x + t)u(t)dt. (4.9) u(x) = −2 − 3x + 0

It is clear that λ = 1, |K(x, t)| 4 and (b − a) = 1. This gives |λ|M (b − a) = 4 < 1.

(4.10)

However, the Fredholm equation (4.9) has an exact solution given by u(x) = 6x. (4.11) A variety of analytic and numerical methods have been used to handle Fredholm integral equations. The direct computation method, the successive approximations method, and converting Fredholm equation to an equivalent boundary value problem are among many traditional methods that were commonly used. However, in this text we will apply the recently developed methods, namely, the Adomian decomposition method (ADM), the modiﬁed decomposition method (mADM), and the variational iteration method (VIM) to handle the Fredholm integral equations. Some of the traditional

4.2 Fredholm Integral Equations of the Second Kind

121

methods, namely, successive approximations method, and the direct computation method will be employed as well. The emphasis in this text will be on the use of these methods rather than proving theoretical concepts of convergence and existence. The theorems of uniqueness, existence, and convergence are important and can be found in the literature. The concern will be on the determination of the solution u(x) of the Fredholm integral equations of the ﬁrst kind and the second kind.

4.2 Fredholm Integral Equations of the Second Kind We will ﬁrst study Fredholm integral equations of the second kind given by b K(x, t)u(t)dt. (4.12) u(x) = f (x) + λ a

The unknown function u(x), that will be determined, occurs inside and outside the integral sign. The kernel K(x, t) and the function f (x) are given real-valued functions, and λ is a parameter. In what follows we will present the methods, new and traditional, that will be used to handle the Fredholm integral equations (4.12).

4.2.1 The Adomian Decomposition Method The Adomian decomposition method (ADM) was introduced and developed by George Adomian in [2–5] and was used before in Chapter 3. The Adomian method will be brieﬂy outlined. The Adomian decomposition method consists of decomposing the unknown function u(x) of any equation into a sum of an inﬁnite number of components deﬁned by the decomposition series ∞ u(x) = un (x), (4.13) n=0

or equivalently u(x) = u0 (x) + u1 (x) + u2 (x) + · · ·

(4.14)

where the components un(x), n 0 will be determined recurrently. The Adomian decomposition method concerns itself with ﬁnding the components u0 , u1 , u2 , . . . individually. As we have seen before, the determination of these components can be achieved in an easy way through a recurrence relation that usually involves simple integrals that can be easily evaluated. To establish the recurrence relation, we substitute (4.13) into the Fredholm integral equation (4.12) to obtain

122

4 Fredholm Integral Equations ∞

b

un (x) = f (x) + λ

K(x, t) a

n=0

or equivalently

∞

un (t) dt,

(4.15)

n=0

b

u0 (x) + u1 (x) + u2 (x) + · · · = f (x) + λ

K(x, t) [u0 (t) + u1 (t) + · · · ] dt. a

(4.16) The zeroth component u0 (x) is identiﬁed by all terms that are not included under the integral sign. This means that the components uj (x), j 0 of the unknown function u(x) are completely determined by setting the recurrence relation b K(x, t)un (t)dt, n 0, (4.17) u0 (x) = f (x), un+1 (x) = λ a

or equivalently u0 (x) = f (x), b K(x, t)u0 (t)dt, u1 (x) = λ a

b

u2 (x) = λ

K(x, t)u1 (t)dt,

(4.18)

a

u3 (x) = λ

b

K(x, t)u2 (t)dt, a

and so on for other components. In view of (4.18), the components u0 (x), u1 (x), u2 (x), u3 (x), . . . are completely determined. As a result, the solution u(x) of the Fredholm integral equation (4.12) is readily obtained in a series form by using the series assumption in (4.13). It is clearly seen that the decomposition method converted the integral equation into an elegant determination of computable components. It was formally shown that if an exact solution exists for the problem, then the obtained series converges very rapidly to that exact solution. The convergence concept of the decomposition series was thoroughly investigated by many researchers to conﬁrm the rapid convergence of the resulting series. However, for concrete problems, where a closed form solution is not obtainable, a truncated number of terms is usually used for numerical purposes. The more components we use the higher accuracy we obtain. Example 4.1 Solve the following Fredholm integral equation 1 u(x) = ex − x + x tu(t)dt.

(4.19)

0

The Adomian decomposition method assumes that the solution u(x) has a series form given in (4.13). Substituting the decomposition series (4.13) into

4.2 Fredholm Integral Equations of the Second Kind

both sides of (4.19) gives ∞ x un (x) = e − x + x

1

t

0

n=0

or equivalently

u0 (x) + u1 (x) + u2 (x) + · · · = e − x + x

un(t)dt,

(4.20)

n=0

1

x

∞

123

t [u0 (t) + u1 (t) + u2 (t) + · · · ] dt.

0

(4.21) We identify the zeroth component by all terms that are not included under the integral sign. Therefore, we obtain the following recurrence relation 1 x u0 (x) = e − x, uk+1 (x) = x tuk (t)dt, k 0. (4.22) 0

Consequently, we obtain u0 (x) = ex − x, 1 u1 (x) = x tu0 (t)dt = x 0

u2 (x) = x u3 (x) = x

tu1 (t)dt = x

0

1

tu2 (t)dt = x

0

1

1

1 0 1 0 1 0

t(et − t)dt =

2 x, 3

2 2 2 t dt = x, 3 9

(4.23)

2 2 2 t dt = x, 9 27

1

2 2 2 t dt = x, 27 81 0 0 and so on. Using (4.13) gives the series solution 2 1 1 1 x u(x) = e − x + x 1 + + + + ··· . (4.24) 3 3 9 27 Notice that the inﬁnite geometric series at the right side has a1 = 1, and the ratio r = 13 . The sum of the inﬁnite series is therefore given by 3 1 S= (4.25) 1 = . 2 1− 3 u4 (x) = x

tu3 (t)dt = x

The series solution (4.24) converges to the closed form solution u(x) = ex ,

(4.26)

obtained upon using (4.25) into (4.24). Example 4.2 Solve the following Fredholm integral equation π2 u(t)dt. u(x) = sin x − x + x

(4.27)

0

Proceeding as before, we substitute the decomposition series (4.13) into both sides of (4.27) to ﬁnd

124

4 Fredholm Integral Equations ∞

un (x) = sin x − x + x

0

n=0

or equivalently u0 (x)+ u1 (x)+ u2 (x)+ · · · = sin x− x+ x

π 2

0

π 2

∞

un (t)dt,

(4.28)

n=0

[u0 (t) + u1 (t) + · · · ] dt. (4.29)

We identify the zeroth component by all terms that are not included under the integral sign. Therefore, we obtain the following recurrence relation: π2 u0 (x) = sin x − x, uk+1 (x) = x uk (t)dt, k 0. (4.30) 0

Consequently, we obtain u0 (x) = sin x − x, π2 π2 x, u0 (t)dt = x − u1 (x) = x 8 0 π2 π2 π4 u1 (t)dt = u2 (x) = x x− x, 8 64 (4.31) 0 π2 4 6 π π x− x, u2 (t)dt = u3 (x) = x 64 512 0 π2 π8 π6 x− x, u3 (t)dt = u4 (x) = x 512 4096 0 and so on. Using (4.13) gives the series solution 2 π2 π π4 u(x) = sin x − x + 1 − x+ − x 8 8 64 6 4 π6 π π8 π − x+ − x + ··· . (4.32) + 64 512 512 4096 We can easily observe the appearance of the noise terms, i.e the identical terms with opposite signs. Canceling these noise terms in (4.32) gives the exact solution u(x) = sin x. (4.33) Example 4.3 Solve the following Fredholm integral equation 1 4 x tu(t)dt. (4.34) u(x) = x + e − + 3 0 Substituting the decomposition series (4.13) into both sides of (4.34) gives 1 ∞ ∞ 4 x un (x) = x + e − + t un (t)dt, (4.35) 3 0 n=0 n=0 or equivalently

4.2 Fredholm Integral Equations of the Second Kind

125

1 4 u0 (x)+u1 (x)+u2 (x)+· · · = x+ex − + t [u0 (t) + u1 (t) + · · · ] dt. (4.36) 3 0 Proceeding as before, we set the following recurrence relation 1 4 x tuk (t)dt, k 0. (4.37) u0 (x) = x + e − , uk+1 (x) = 3 0 Consequently, we obtain 1 4 2 tu0 (t)dt = , u0 (x) = x + ex − , u1 (x) = 3 3 0 1 1 1 1 (4.38) u3 (x) = tu1 (t)dt = , tu2 (t)dt = , u2 (x) = 3 6 0 0 1 1 u4 (x) = , tu3 (t)dt = 12 0 and so on. Using (4.13) gives the series solution 1 1 1 4 2 x 1 + + + + ··· . (4.39) u(x) = x + e − + 3 3 2 4 8 Notice that the inﬁnite geometric series at the right side has a1 = 1, and the ratio r = 12 . The sum of the inﬁnite series is therefore given by 1 = 2. (4.40) S= 1 − 12 The series solution (4.39) converges to the closed form solution u(x) = x + ex . (4.41) Example 4.4 Solve the following Fredholm integral equation π tu(t)dt. u(x) = 2 + cos x +

(4.42)

0

Proceeding as before we ﬁnd ∞ un (x) = 2 + cos x +

u0 (x) + u1 (x) + u2 (x) + · · · = 2 + cos x +

t

0

n=0

or equivalently

π

0

∞

un (t)dt,

π

t [u0 (t) + u1 (t) + · · · ] dt. (4.44)

We next set the following recurrence relation π u0 (x) = 2 + cos x, uk+1 (x) = tuk (t)dt, 0

This in turn gives

(4.43)

n=0

k 0.

(4.45)

126

4 Fredholm Integral Equations

u0 (x) = 2 + cos x, π u0 (t)dt = −2 + π 2 , u1 (x) = u2 (x) = u3 (x) =

0

π

1 u1 (t)dt = −π 2 + π 4 , 2

π

1 1 u2 (t)dt = − π 4 + π 6 , 2 4

0

0

(4.46)

π

1 1 u3 (t)dt = − π 6 + π 8 , 4 8 0 and so on. Using (4.13) gives the series solution 1 4 2 2 u(x) = 2 + cos x + (−2 + π ) + −π + π 2 1 1 1 1 (4.47) + − π4 + π6 + − π6 + π8 + · · · . 2 4 4 8 We can easily observe the appearance of the noise terms, i.e the identical terms with opposite signs. Canceling these noise terms in (4.47) gives the exact solution u(x) = cos x. (4.48) u4 (x) =

Example 4.5 Solve the following Fredholm integral equation π 1 4 sec2 xu(t)dt. (4.49) u(x) = 1 + 2 0 Substituting the decomposition series (4.13) into both sides of (4.49) gives π4 ∞ ∞ 1 un (x) = 1 + sec2 x un (t)dt, (4.50) 2 0 n=0 n=0 or equivalently

π4 1 [u0 (t) + u1 (t) + · · · ] dt. (4.51) u0 (x)+ u1 (x)+ u2 (x)+ · · · = 1 + sec2 x 2 0 Proceeding as before, we set the recurrence relation π4 1 2 u0 (x) = 1, uk+1 (x) = sec x uk (t)dt, k 0. (4.52) 2 0 Consequently, we obtain u0 (x) = 1, π4 1 π 2 u0 (t)dt = sec2 x, u1 (x) = sec x 2 8 0

4.2 Fredholm Integral Equations of the Second Kind

u2 (x) =

1 sec2 x 2

1 u3 (x) = sec2 x 2 1 u4 (x) = sec2 x 2

π 4

0 π 4

0 π 4

0

127

u1 (t)dt =

π sec2 x, 16

u2 (t)dt =

π sec2 x, 32

u3 (t)dt =

π sec2 x, 64

(4.53)

and so on. Using (4.13) gives the series solution 1 1 1 π 2 u(x) = 1 + sec x 1 + + + + · · · . (4.54) 8 2 4 8 The sum of the inﬁnite series at the right side is S = 2. The series solution (4.54) converges to the closed form solution π (4.55) u(x) = 1 + sec2 x. 4 Example 4.6 Solve the following Fredholm integral equation π u(x) = πx + sin 2x + x tu(t)dt.

(4.56)

−π

Proceeding as before we ﬁnd ∞ un (x) = πx + sin 2x + x n=0

π

t −π

∞

un (t)dt.

(4.57)

n=0

To determine the components of u(x), we use the recurrence relation π tuk (t)dt, k 0. (4.58) u0 (x) = πx + sin 2x, uk+1 (x) = x −π

This in turn gives u0 (x) = πx + sin 2x, π 2 u0 (t)dt = −πx + π 4 x, u1 (x) = x 3 0 π 2 4 u2 (x) = x u1 (t)dt = − π 4 x + π 7 x, 3 9 0 π 4 8 u2 (t)dt = − π 7 x + π 10 x, u3 (x) = x 9 27 0 and so on. Using (4.13) gives the series solution 2 4 2 u(x) = πx + sin 2x + −π + π 4 x + − π 4 + π7 x 3 3 9 8 4 + − π 7 + π10 x + · · · . 9 27 Canceling the noise terms in (4.60) gives the exact solution u(x) = sin 2x.

(4.59)

(4.60)

(4.61)

128

4 Fredholm Integral Equations

Exercises 4.2.1 In Exercises 1–20, solve the following Fredholm integral equations by using the Adomian decomposition method 1 1 1. u(x) = ex + 1 − e + u(t)dt 2. u(x) = ex + e−1 u(t)dt 3. u(x) = cos x + 2x + 5. u(x) = ex+2 − 2

0

0

xtu(t)dt

0

1

ex+t u(t)dt

0

7. u(x) = x + (1 − x)ex + 8. u(x) = 1 +

1 sin2 x 2

9. u(x) = xex −

π 2

1 1 + 2 2

0

u(t)dt

1 2

19 2 x + 15

π 2

0

16. u(x) = −x + sin x +

6. u(x) = ex +

π 2

0

xtu(t)dt

ex+1 − 1 − x+1

1

0

ext u(t)dt

x2 et(x−1) u(t)dt

1

0

13. u(x) = x + sin x −

1

4. u(x) = sin x − x +

u(t)dt

0

11. u(x) = x cos x + 1 +

15. u(x) = 1 −

π

12. u(x) = sin x + 14. u(x) = 1 −

1 1 + 2 2

0

π 2

0

π 2

u(t)dt

sin x cos tu(t)dt

1 2 x + 15

1 −1

(xt + x2 t2 )u(t)dt

(xt + x2 t2 )u(t)dt

π 2

0

u(t)dt

xu(t)dt

−1

π

0

1

10. u(x) = x sin x −

(1 + x − t)u(t)dt

1 e−(x+1) − 1 e−xt u(t)dt + x+1 0 1 3 1 18. u(x) = ex − ex+2 + ex+t u(t)dt 2 2 0 π π π 1 2 4 cos x sin tu(t)dt 20. u(x) = − sec2 x − u(t)dt 19. u(x) = cos x + 2 4 0 0

17. u(x) = e−x +

4.2.2 The Modiﬁed Decomposition Method As stated before, the Adomian decomposition method provides the solutions in an inﬁnite series of components. The components uj , j 0 are easily computed if the inhomogeneous term f (x) in the Fredholm integral equation b u(x) = f (x) + λ K(x, t)u(t)dt, (4.62) a

consists of a polynomial of one or two terms. However, if the function f (x) consists of a combination of two or more of polynomials, trigonometric func-

4.2 Fredholm Integral Equations of the Second Kind

129

tions, hyperbolic functions, and others, the evaluation of the components uj , j 0 requires more work. A reliable modiﬁcation of the Adomian decomposition method was presented and used in Chapter 3, and it was shown that this modiﬁcation facilitates the computational work and accelerates the convergence of the series solution. As presented before, the modiﬁed decomposition method depends mainly on splitting the function f (x) into two parts, therefore it cannot be used if the f (x) consists of only one term. The modiﬁed decomposition method will be brieﬂy outlined here, but will be used in this section and in other chapters as well. The standard Adomian decomposition method employs the recurrence relation u0 (x) = f (x), b (4.63) uk+1 (x) = λ K(x, t)uk (t)dt, k 0, a

where the solution u(x) is expressed by an inﬁnite sum of components deﬁned by ∞ un (x). (4.64) u(x) = n=0

In view of (4.63), the components un (x), n 0 are readily obtained. The modiﬁed decomposition method presents a slight variation to the recurrence relation (4.63) to determine the components of u(x) in an easier and faster manner. For many cases, the function f (x) can be set as the sum of two partial functions, namely f1 (x) and f2 (x). In other words, we can set (4.65) f (x) = f1 (x) + f2 (x). In view of (4.65), we introduce a qualitative change in the formation of the recurrence relation (4.63). The modiﬁed decomposition method identiﬁes the zeroth component u0 (x) by one part of f (x), namely f1 (x) or f2 (x). The other part of f (x) can be added to the component u1 (x) that exists in the standard recurrence relation. The modiﬁed decomposition method admits the use of the modiﬁed recurrence relation: u0 (x) = f1 (x), b K(x, t)u0 (t)dt, u1 (x) = f2 (x) + λ (4.66) a b uk+1 (x) = λ K(x, t)uk (t)dt, k 1. a

It is obvious that the diﬀerence between the standard recurrence relation (4.63) and the modiﬁed recurrence relation (4.66) rests only in the formation of the ﬁrst two components u0 (x) and u1 (x) only. The other components uj , j 2 remain the same in the two recurrence relations. Although this variation in the formation of u0 (x) and u1 (x) is slight, however it has been shown that it accelerates the convergence of the solution and minimizes the

130

4 Fredholm Integral Equations

size of calculations. Moreover, reducing the number of terms in f1 (x) aﬀects not only the component u1 (x), but also the other components as well. We here emphasize on the two important remarks made in Chapter 3. First, by proper selection of the functions f1 (x) and f2 (x), the exact solution u(x) may be obtained by using very few iterations, and sometimes by evaluating only two components. The success of this modiﬁcation depends only on the proper choice of f1 (x) and f2 (x), and this can be made through trials only. A rule that may help for the proper choice of f1 (x) and f2 (x) could not be found yet. Second, if f (x) consists of one term only, the modiﬁed decomposition method cannot be used in this case. Example 4.7 Solve the Fredholm integral equation by using the modiﬁed decomposition method 1 1 4x 4 tu(t)dt. (4.67) u(x) = 3x + e − (17 + 3e ) + 16 0 We ﬁrst decompose f (x) given by f (x) = 3x + e4x −

1 (17 + 3e4 ), 16

(4.68)

into two parts, namely 1 (17 + 3e4 ). 16 We next use the modiﬁed recurrence formula (4.66) to obtain u0 (x) = f1 (x) = 3x + e4x , 1 1 tu0 (t)dt = 0, u1 (x) = − (17 + 3e4 ) + 16 0 x uk+1 (x) = K(x, t)uk (t)dt = 0, k 1. f1 (x) = 3x + e4x ,

f2 (x) = −

(4.69)

(4.70)

0

It is obvious that each component of uj , j 1 is zero. This in turn gives the exact solution by u(x) = 3x + e4x . (4.71) Example 4.8 Solve the Fredholm integral equation by using the modiﬁed decomposition method 1 π 1 − 2 sinh earctan t u(t)dt. (4.72) u(x) = + 1 + x2 4 −1 Proceeding as before we split f (x) given by 1 π f (x) = − 2 sinh , 2 1+x 4 into two parts, namely

(4.73)

4.2 Fredholm Integral Equations of the Second Kind

1 π , f2 (x) = −2 sinh . 1 + x2 4 We next use the modiﬁed recurrence formula (4.66) to obtain 1 , u0 (x) = f1 (x) = 1 + x2 1 π u1 (x) = −2 sinh + earctan t u0 (t)dt = 0, 4 −1 1 earctan t uk (t)dt = 0, k 1. uk+1 (x) = f1 (x) =

131

(4.74)

(4.75)

−1

It is obvious that each component of uj , j 1 is zero. This in turn gives the exact solution by 1 u(x) = . (4.76) 1 + x2 Example 4.9 Solve the Fredholm integral equation by using the modiﬁed decomposition method 2−π 2 1 2 1 −1 x + 1 u(t)dt. (4.77) u(x) = x + sin + x + x 2 2 2 −1 We decompose f (x) given by f (x) = x + sin−1

x+1 2−π 2 + x , 2 2

(4.78)

into two parts given by 2−π 2 x+1 , f2 (x) = x . 2 2 We next use the modiﬁed recurrence formula (4.66) to obtain x+1 u0 (x) = x + sin−1 , 2 2−π 2 1 2 1 x + x u0 (t)dt = 0, u1 (x) = 2 2 −1 1 uk (t)dt = 0, k 1. uk+1 (x) = − f1 (x) = x + sin−1

(4.79)

(4.80)

−1

It is obvious that each component of uj , j 1 is zero. The exact solution is therefore given by x+1 . (4.81) u(x) = x + sin−1 2 Example 4.10 Solve the Fredholm integral equation by using the modiﬁed decomposition method

132

4 Fredholm Integral Equations

u(x) = sec2 x + x2 + x −

π 4

0

4 2 x + x u(t) dt. π

(4.82)

The function f (x) consists of three terms. By trial we split f (x) given by f (x) = sec2 x + x2 + x, into two parts

f1 (x) = sec2 x,

f2 (x) = x2 + x.

Using the modiﬁed recurrence formula (4.66) gives u0 (x) = f1 (x) = sec2 x, π4 4 2 2 x + xu0 (t) dt = 0, u1 (x) = x + x − π 0 π4 K(x, t)uk (t)dt = 0, k 1. uk+1 (x) =

(4.83) (4.84)

(4.85)

0

As a result, the exact solution is given by u(x) = sec2 x.

(4.86)

Exercises 4.2.2 Use the modiﬁed decomposition method to solve the following Fredholm integral equations: π 2 1. u(x) = sin x − x + x tu(t)dt 0

2. u(x) = (π + 2)x + sin x − cos x − x 3. u(x) = ex + 12x2 + (3 + e)x − 4 −

π

tu(t)dt

0 1

0

(x − t)u(t)dt

1 x+1 x−1 − sin−1 − xu(t)dt 2 2 0 1 (x4 − t4 )u(t)dt 5. u(x) = −6 + 14x4 + 21x2 + x − 4. u(x) = (π − 2)x + sin−1

−1

6. u(x) = x + x4 + 9ex+1 − 23ex − 7. u(x) = x + ex − 2ex−1 + 2ex − 8. u(x) = xex +

xex+1 + 1 − (x + 1)2

1 0

1

0

1

0

ex+t u(t)dt

ex−t u(t)dt

ext u(t)dt

9. u(x) = ex+1 + ex−1 + (e2 + 1)ex−1 − 10. u(x) =

7 2 + x + x2 + x3 − 15 12

1

0

1

0

ex−t u(t)dt

(1 + x − t)u(t)dt

4.2 Fredholm Integral Equations of the Second Kind 11. u(x) = e2x −

1 2 (e + 1)x + 4

1

12. u(x) = ex (e 3 − 1) + e2x − 13. u(x) = x + ex − xex +

1

0

1 0

1 3

133

xtu(t)dt

1

0

5

ex− 3 t u(t)dt

x2 et(x−1) u(t)dt

14. u(x) = (π 2 + 2π − 4) − (π + 2)x + x(sin x − cos x) +

π 0

(x − t)u(t)dt

1 (π − 2)(x + 1) + x tan−1 x − (1 + x − t)u(t)dt 2 −1 1 1+e ex − xu(t)dt 16. u(x) = ln x+ 2 1 + ex 0 15. u(x) =

4.2.3 The Noise Terms Phenomenon It was shown that a proper selection of f1 (x) and f2 (x) is essential to use the modiﬁed decomposition method. However, the noise terms phenomenon, that was introduced in Chapter 3, demonstrated a fast convergence of the solution. This phenomenon was presented before, therefore we present here the main steps for using this eﬀect concept. The noise terms as deﬁned before are the identical terms with opposite signs that may appear between components u0 (x) and u1 (x). Other noise terms may appear between other components. By canceling the noise terms between u0 (x) and u1 (x), even though u1 (x) contains further terms, the remaining non-canceled terms of u0 (x) may give the exact solution of the integral equation. The appearance of the noise terms between u0 (x) and u1 (x) is not always suﬃcient to obtain the exact solution by canceling these noise terms. Therefore, it is necessary to show that the non-canceled terms of u0 (x) satisfy the given integral equation. It was formally proved in [6] that a necessary condition for the appearance of the noise terms is required. The conclusion made in [6] is that the zeroth component u0 (x) must contain the exact solution u(x) among other terms. The phenomenon of the useful noise terms will be explained by the following illustrative examples. Example 4.11 Solve the Fredholm integral equation by using the noise terms phenomenon: π2 u(x) = x sin x − x + xu(t)dt. (4.87) 0

Following the standard Adomian method we set the recurrence relation

134

4 Fredholm Integral Equations

u0 (x) = x sin x − x, π2 xuk (t)dt, uk+1 (x) = 0

This gives

k 0.

u0 (x) = x sin x − x, π2 π2 x. xu0 (t)dt = x − u1 (x) = 8 0

(4.88)

(4.89)

The noise terms ∓x appear in u0 (x) and u1 (x). Canceling this term from the zeroth component u0 (x) gives the exact solution u(x) = x sin x, (4.90) that justiﬁes the integral equation. The other terms of u1 (x) vanish in the limit with other terms of the other components. Example 4.12 Solve the Fredholm integral equation by using the noise terms phenomenon: π2 π u(x) = sin x + cos x − x + xtu(t)dt. (4.91) 2 0 The standard Adomian method gives the recurrence relation π u0 (x) = sin x + cos x − x, 2 π2 (4.92) xtuk (t)dt, k 0. uk+1 (x) = 0

This gives

π u0 (x) = sin x + cos x − x, 2 π2 π π4 xtu0 (t)dt = x − u1 (x) = x. 2 48 0

(4.93)

The noise terms ∓ π2 x appear in u0 (x) and u1 (x). Canceling this term from the zeroth component u0 (x) gives the exact solution u(x) = sin x + cos x, (4.94) that justiﬁes the integral equation. It is to be noted that the other terms of u1 (x) vanish in the limit with other terms of the other components. Example 4.13 Solve the Fredholm integral equation by using the noise terms phenomenon: π2 sin x π3 2 u(t)dt. (4.95) u(x) = x + + x + x ln 2 − x 1 + cos x 24 0 The standard Adomian method gives the recurrence relation:

4.2 Fredholm Integral Equations of the Second Kind

π3 sin x + x + x ln 2, u0 (x) = x2 + 1 + cos x 24 π2 u(t)dt, k 0. uk+1 (x) = −x

135

(4.96)

0

This gives sin x π3 u0 (x) = x2 + + x + x ln 2, 1 + cos x 24 π2 π2 ln 2 π5 π3 x− x. xu0 (t)dt = − x − x ln 2 − u1 (x) = − 24 8 192 0

(4.97)

3

The noise terms ± π24 x, ±x ln 2 appear in u0 (x) and u1 (x). Canceling these terms from the zeroth component u0 (x) gives the exact solution sin x u(x) = x2 + , (4.98) 1 + cos x that justiﬁes the integral equation. The other terms of u1 (x) vanish in the limit with other terms of the other components. Example 4.14 Solve the Fredholm integral equation by using the noise terms phenomenon π π3 2 u(x) = x + x cos x + x − 2x − x u(t)dt. (4.99) 3 0 Proceeding as before, we set the recurrence relation π3 x − 2x, u0 (x) = x2 + x cos x + 3 π (4.100) uk (t)dt, k 0. uk+1 (x) = −x 0

This gives π3 u0 (x) = x2 + x cos x + x − 2x, 3 π π3 π5 u0 (t)dt = −x −2 − π 2 + u1 (x) = −x + . 3 6 0

(4.101)

3

The noise terms ± π3 x, ∓2x appear in u0 (x) and u1 (x). Canceling these terms from the zeroth component u0 (x) gives the exact solution u(x) = x2 + x cos x, (4.102) that justiﬁes the integral equation (4.99).

Exercises 4.2.3 Use the noise terms phenomenon to solve the following Fredholm integral equations: π π cos x 2 xu(t)dt + ln 2 x + − 1. u(x) = 1 + 2 1 + sin x 0

136

4 Fredholm Integral Equations

π π2 2. u(x) = x 1 + π + xu(t)dt + x sin x − 2 0 π π3 4 xu(t)dt 3. u(x) = x 1 + + x2 + sec2 x − 192 0 π xu(t)dt 4. u(x) = x(π − 2 + sin x + cos x) −

+ ln 2 +

0

π sin x 2 − xu(t)dt 8 1 + cos x 0 π cos x 2 xu(t)dt 6. u(x) = x(1 + ln 2) + sin x + − 1 + sin x 0 π sin x(2 + sin x) π 2 − 7. u(x) = x + xu(t)dt 2 1 + sin x 0 π sin x π 2 −1 x+ − 8. u(x) = xu(t)dt 2 1 + sin x 0 √ π π− 3 cos x 3 xu(t)dt 9. u(x) = x+ − 3 1 + sin x 0 π π2 2 xu(t)dt 10. u(x) = 4 − x + x2 (sin x + cos x) + 2 −π 2 π 1 4 xu(t)dt 11. u(x) = x + sin(2x) − 4 0 π π−2 4 xu(t)dt 12. u(x) = x + x cos(2x) − 8 0 1 π2 1 − x tan−1 tu(t)dt 13. u(x) = x + 1 + x2 32 0 1 14. u(x) = πx + cos−1 x − xu(t)dt 5. u(x) = x 1 +

π2

−1

1 π 15. u(x) = − x + x cos−1 x − xu(t)dt 4 −1 1 π−2 x + x tan−1 x − 16. u(x) = xu(t)dt 2 −1

4.2.4 The Variational Iteration Method In Chapter 3, the variational iteration method was used to handle Volterra integral equations, where the Volterra integral equation was converted to an initial value problem or to an equivalent integro-diﬀerential equation. The method provides rapidly convergent successive approximations to the exact solution if such a closed form solution exists. In this section, we will apply the variational iteration method to handle Fredholm integral equation. The method works eﬀectively if the kernel

4.2 Fredholm Integral Equations of the Second Kind

137

K(x, t) is separable and can be written in the form K(x, t) = g(x)h(t). The approach to be used here is identical to the approach used in the previous chapter. This means that we should diﬀerentiate both sides of the Fredholm integral equation to convert it to an identical Fredholm integro-diﬀerential equation. It is important to note that integro-diﬀerential equation needs an initial condition that should be deﬁned. In view of this fact, we will study only the cases where g(x) = xn , n 1. Solving Fredholm integro-diﬀerential equation by the variational iteration method will be studied again in details in Chapter 6. The standard Fredholm integral equation is of the form b K(x, t)u(t)dt, (4.103) u(x) = f (x) + a

or equivalently

b

u(x) = f (x) + g(x)

h(t)u(t)dt.

(4.104)

a

Recall that the integral at the right side represents a constant value. Diﬀerentiating both sides of (4.104) with respect to x gives b u (x) = f (x) + g (x) h(t)u(t)dt. (4.105) a

The correction functional for the integro-diﬀerential equation (4.105) is x b λ(ξ) un (ξ) − f (ξ) − g (ξ) h(r)˜ un (r)dr dξ. un+1 (x) = un (x) + 0

a

(4.106) As presented before, the variational iteration method is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ(ξ) that can be identiﬁed optimally via integration by parts and by using a restricted variation. However, λ(ξ) = −1 for ﬁrst order integro-diﬀerential equations. Having determined λ, an iteration formula, without restricted variation, given by x b h(r)un (r)dr dξ, (4.107) un+1 (x) = un (x) − un (ξ) − f (ξ) − g (ξ) 0

a

is used for the determination of the successive approximations un+1 (x), n 0 of the solution u(x). The zeroth approximation u0 can be any selective function. However, using the given initial value u(0) is preferably used for the selective zeroth approximation u0 as will be seen later. Consequently, the solution is given by (4.108) u(x) = lim un (x). n→∞

138

4 Fredholm Integral Equations

The variational iteration method will be illustrated by studying the following Fredholm integral equations. Example 4.15 Use the variational iteration method to solve the Fredholm integral equation 1 u(x) = ex − x + x tu(t)dt. (4.109) 0

Diﬀerentiating both sides of this equation with respect to x yields 1 tu(t)dt. u (x) = ex − 1 +

(4.110)

The correction functional for this equation is given by x 1 ξ run (r)dr dξ, un+1 (x) = un (x) − un (ξ) − e + 1 −

(4.111)

0

0

0

where we used λ = −1 for ﬁrst-order integro-diﬀerential equations. Notice that the initial condition u(0) = 1 is obtained by substituting x = 0 into (4.109). We can use the initial condition to select u0 (x) = u(0) = 1. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1,

u0 (ξ) − eξ + 1 −

1 ru0 (r)dr dξ = ex − x, 2 0 0 x 1 1 x, u2 (x) = u1 (x) − ru1 (r)dr dξ = ex − u1 (ξ) − eξ + 1 − 2 × 3 0 0 x 1 1 ξ u3 (x) = u2 (x) − ru2 (r)dr dξ = ex − x, u2 (ξ) − e + 1 − 2 × 32 0 0 x 1 1 u3 (ξ) − eξ + 1 − u4 (x) = u3 (x) − ru3 (r)dr dξ = ex − x, 2 × 33 0 0 .. . 1 x, n 0. (4.112) un+1 = ex − 2 × 3n The VIM admits the use of u(x) = lim un (x) = ex . (4.113) u1 (x) = u0 (x) −

x

1

n→∞

Example 4.16 Use the variational iteration method to solve the Fredholm integral equation π2 u(t)dt. (4.114) u(x) = sin x − x + x 0

Diﬀerentiating both sides of this equation with respect to x gives

4.2 Fredholm Integral Equations of the Second Kind

u (x) = cos x − 1 +

π 2

139

u(t)dt.

(4.115)

0

The correction functional for this equation is given by x π2 un+1 (x) = un (x) − un (ξ) − cos ξ + 1 − un (r)dr dξ, 0

(4.116)

0

where we used λ = −1 for ﬁrst-order integro-diﬀerential equations. The initial condition u(0) = 0 is obtained by substituting x = 0 into (4.114). We can use the initial condition to select u0 (x) = u(0) = 0. Using this selection into the correction functional gives the following successive approximations u0 (x) = 0, x π2 u0 (ξ) − cos ξ + 1 − u1 (x) = u0 (x) − u0 (r)dr dξ = sin x − x, u2 (x) = u1 (x) −

0

0

x

u1 (ξ)

π 2

− cos ξ + 1 − u1 (r)dr dξ 0 π2 x , = (sin x − x) + x − 8 π 0

x

(4.117)

2

u2 (ξ) − cos ξ + 1 − u2 (r)dr dξ 0 0 2 π2 π π4 = (sin x − x) + x − x + x− x , 8 8 64 and so on. Canceling the noise terms, the exact solution is given by u3 (x) = u2 (x) −

u(x) = sin x.

(4.118)

Example 4.17 Use the variational iteration method to solve the Fredholm integral equation π xu(t)dt. (4.119) u(x) = −2x + sin x + cos x + 0

Diﬀerentiating both sides of this equation with respect to x gives π u (x) = −2 + cos x − sin x + u(t)dt.

(4.120)

0

The correction functional for this equation is given by x un (ξ) + 2 − cos ξ + sin ξ − un+1 (x) = un (x) − 0

0

π

un (r)dr dξ.

(4.121) The initial condition u(0) = 1 is obtained by substituting x = 0 into (4.119). Using this selection into the correction functional gives the following successive approximations

140

u0 (x) = 1, u1 (x) = u0 (x) −

4 Fredholm Integral Equations x

0

u0 (ξ)

+ 2 − cos ξ + sin ξ −

π

0

u0 (r)dr dξ

= sin x + cos x + (πx − 2x), x π u1 (r)dr dξ u1 (ξ) + 2 − cos ξ + sin ξ − u2 (x) = u1 (x) − 0 0 π3 x , (4.122) = sin x + cos x + (πx − 2x) + −πx + 2x − π 2 x + 2 x π u3 (x) = u2 (x) − u2 (r)dr dξ u2 (ξ) + 2 − cos ξ + sin ξ − 0 0 π3 x = sin x + cos x + (πx − 2x) + −πx + 2x − π 2 x + 2 3 π x + ··· , + π2 x − 2 and so on. Canceling the noise terms, the exact solution is given by u(x) = sin x + cos x. (4.123) Example 4.18 Use the variational iteration method to solve the Fredholm integral equation π2 u(x) = −x3 + cos x + x3 u(t)dt. (4.124) 0

Diﬀerentiating both sides of this equation with respect to x gives π2 u (x) = −3x2 − sin x + 3x2 u(t)dt.

(4.125)

0

The correction functional for this equation is given by π2 x 2 2 un (r)dr dξ. (4.126) un (ξ) + 3ξ + sin ξ − 3ξ un+1 (x) = un (x)− 0

0

The initial condition is u(0) = 1. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1, π x

2

u0 (r)dr dξ u0 (ξ) + 3ξ 2 + sin ξ − 3ξ 2 u1 (x) = u0 (x) − 0 0 π = cos x + x3 − x3 , 2 π x

2

u1 (r)dr dξ u1 (ξ) + 3ξ 2 + sin ξ − 3ξ 2 0 π π 3 π4 3 π5 3 3 3 3 = cos x + x −x + − x +x − x + x , 2 2 64 128 and so on. Canceling the noise terms, the exact solution is given by u2 (x) = u1 (x) −

0

(4.127)

4.2 Fredholm Integral Equations of the Second Kind

141

u(x) = cos x.

(4.128)

Exercises 4.2.4 Solve the following Fredholm integral equations by using the the variational iteration method π π 2 1. u(x) = cos x + 2x + xtu(t)dt 2. u(x) = sin x − x + xtu(t)dt 1 2 x + 3. u(x) = 1 − 15 4. u(x) = 1 −

19 2 x + 15

0

1

−1

1

−1

0

(xt + x2 t2 )u(t)dt (xt + x2 t2 )u(t)dt

5. u(x) = (π + 2)x + sin x − cos x − x 6. u(x) = e2x −

1 2 (e + 1)x + 4

7. u(x) = 1 + 9x + 2x2 + x3 −

π

0

tu(t)dt

1

xtu(t)dt

0

1 0

(20xt + 10x2 t2 )u(t)dt

π π x + sin x − cos x + x cos tu(t)dt 2 0 2 1 3 1 xtu(t)dt 10. u(x) = 2x + ex − xtu(t)dt 9. u(x) = 1 + x + ex − 3 0 4 0 π 1 1 4 + sin(2x) − 11. u(x) = x xu(t)dt 12. u(x) = ex + 2xe−1 − xtu(t)dt 4 −1 0 3 π π π3 π 2 xtu(t)dt 14. u(x) = xtu(t)dt + 3 x − cos x − x − sin x − 13. u(x) = 3 24 0 0 π √ 4 x sec tu(t)dt 15. u(x) = 2 x + sec x + tan x − 8. u(x) =

π 16. u(x) = x + sin x + cos x − 8

0

0

π 4

x sin tu(t)dt

4.2.5 The Direct Computation Method In this section, the direct computation method will be applied to solve the Fredholm integral equations. The method approaches Fredholm integral equations in a direct manner and gives the solution in an exact form and not in a series form. It is important to point out that this method will be applied for the degenerate or separable kernels of the form n gk (x)hk (t). (4.129) K(x, t) = k=1

142

4 Fredholm Integral Equations

Examples of separable kernels are x − t, xt, x2 − t2 , xt2 + x2 t, etc. The direct computation method can be applied as follows: 1. We ﬁrst substitute (4.129) into the Fredholm integral equation the form b u(x) = f (x) + K(x, t) u(t)dt. (4.130) a

2. This substitution gives b u(x) = f (x) + g1 (x) h1 (t)u(t)dt + g2 (x) a

b

h2 (t)u(t)dt + · · · a

b

+gn (x)

(4.131)

hn (t)u(t)dt. a

3. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (4.131) becomes u(x) = f (x) + λα1 g1 (x) + λα2 g2 (x) + · · · + λαn gn (x), where

αi =

(4.132)

b

hi (t)u(t)dt,

1 i n.

(4.133)

a

4. Substituting (4.132) into (4.133) gives a system of n algebraic equations that can be solved to determine the constants αi , 1 i n. Using the obtained numerical values of αi into (4.132), the solution u(x) of the Fredholm integral equation (4.130) is readily obtained. Example 4.19 Solve the Fredholm integral equation by using the direct computation method 1 1 2 2 x tu(t)dt. (4.134) u(x) = 3x + 3x + 2 0 The kernel K(x, t) = x2 t is separable. Consequently, we rewrite (4.134) as 1 2 1 2 u(x) = 3x + 3x + x tu(t)dt. (4.135) 2 0 The integral at the right side is equivalent to a constant because it depends only on functions of the variable t with constant limits of integration. Consequently, Equation (4.135) can be rewritten as 1 u(x) = 3x + 3x2 + αx2 , (4.136) 2 where 1 tu(t)dt. (4.137) α= 0

To determine α, we substitute (4.136) into (4.137) to obtain 1 1 α= t 3t + 3t2 + αt2 dt. 2 0

(4.138)

4.2 Fredholm Integral Equations of the Second Kind

143

Integrating the right side of (4.138) yields 7 1 α = + α, 4 8 that gives α = 2.

(4.139)

Substituting (4.140) into (4.136) leads to the exact solution u(x) = 3x + 4x2 , obtained by substituting α = 2 in (4.136).

(4.141)

(4.140)

Example 4.20 Solve the Fredholm integral equation by using the direct computation method π3 1 1 u(x) = x + sec x tan x − x u(t)dt. (4.142) 3 3 0 The integral at the right side is equivalent to a constant because it depends only on functions of the variable t with constant limits of integration. Consequently, Equation (4.142) can be rewritten as 1 1 (4.143) u(x) = x + sec x tan x − αx, 3 3 where π3 u(t)dt. (4.144) α= 0

To determine α, we substitute (4.143) into (4.144) to obtain π3 1 1 α= t + sec t tan t − αt dt. 3 3 0 Integrating the right side of (4.145) yields 1 1 α = 1 + π 2 − απ2 , 54 54 that gives α = 1.

(4.145)

(4.146) (4.147)

Substituting (4.147) into (4.143) gives the exact solution u(x) = sec x tan x.

(4.148)

Example 4.21 Solve the Fredholm integral equation by using the direct computation method 1 (30xt2 + 20x2 t)u(t)dt. (4.149) u(x) = 11x + 10x2 + x3 − 0

The kernel K(x, t) = 30xt2 + 20x2 t is separable. Consequently, we rewrite (4.149) as 1 1 u(x) = 11x + 10x2 + x3 − 30x t2 u(t)dt − 20x2 tu(t)dt. (4.150) 0

0

144

4 Fredholm Integral Equations

Each integral at the right side is equivalent to a constant because it depends only on functions of the variable t with constant limits of integration. Consequently, Equation (4.150) can be rewritten as u(x) = (11 − 30α)x + (10 − 20β)x2 + x3 , (4.151) where

1

α=

t2 u(t)dt,

0

β=

(4.152)

1

tu(t)dt. 0

To determine the constants α and β, we substitute (4.151) into (4.152) to obtain 1 59 15 t2 ((11 − 30α)x + (10 − 20β)x2 + x3 )dt = − α − 4β, α= 12 2 0 (4.153) 1 191 2 3 t((11 − 30α)t + (10 − 20β)t + x )dt = β= − 10α − 5β. 30 0 Solving this system of algebraic equations gives 9 11 , β= . (4.154) α= 30 20 Substituting (4.154) into (4.151) gives the exact solution u(x) = x2 + x3 .

(4.155)

Example 4.22 Solve the Fredholm integral equation by using the direct computation method 1 (1 + 30xt2 + 12x2 t)u(t)dt. (4.156) u(x) = 4 + 45x + 26x2 − 0

The kernel K(x, t) = 1 + 30xt2 + 12x2 t is separable. Consequently, we rewrite (4.156) as 1 1 1 u(t)dt − 30x t2 u(t)dt − 12x2 tu(t)dt. u(x) = 4 + 45x + 26x2 − 0

0

0

(4.157) Each integral at the right side is equivalent to a constant because it depends only on functions of the variable t with constant limits of integration. Consequently, Equation (4.157) can be rewritten as u(x) = (4 − α) + (45 − 30β)x + (26 − 12γ)x2 , (4.158) where

α=

1

u(t)dt, 0

β= 0

1

t2 u(t)dt,

1

γ=

tu(t)dt.

(4.159)

0

To determine the constants α, β and γ, we substitute (4.158) into each equation of (4.159) to obtain

4.2 Fredholm Integral Equations of the Second Kind

1

α=

145

0

(4 − α) + (45 − 30β)t + (26 − 12γ)t2 dt

211 = − α − 15β − 4γ, 6 1

β= t2 (4 − α) + (45 − 30β)t + (26 − 12γ)t2 dt 0

(4.160)

1067 1 15 12 = − α − β − γ, 60 3 2 5 1

γ= t (4 − α) + (45 − 30β)t + (26 − 12γ)t2 dt 0

47 1 = − α − 10β − 3γ. 2 2 Unlike the previous examples, we obtain a system of three equations in three unknowns α, β, and γ. Solving this system of algebraic equations gives 23 43 , γ= . (4.161) α = 3, β = 30 12 Substituting (4.161) into (4.158), the exact solution is given by (4.162) u(x) = 1 + 2x + 3x2 .

Exercises 4.2.5 Use the direct computation method to solve the following Fredholm integral equations: 1 (20xt + 10x2 t2 )u(t)dt 1. u(x) = 1 + 9x + 2x2 + x3 − 0

2. u(x) = −8 + 11x − x2 + x3 − 3. u(x) = −11 + 9x + x3 + x4 − 4. u(x) = −15 + 10x3 + x4 −

5. u(x) = 1 + 7x + 20x2 + x3 −

0

1 0

1

1

0

(12x − 20t)u(t)dt (20x − 30t)u(t)dt

(20x3 − 56t3 )u(t)dt

1 0

(10xt2 + 20x2 t)u(t)dt

π 2 6 √ − 1 x + sec x tan x − xu(t)dt 3 0 π √ 2π 3 − ln(2 + 3) x + sec x tan x − xtu(t)dt 7. u(x) = 3 0 1 ln(xt)u(t)dt, 0 < x 1 8. u(x) = 1 +

6. u(x) =

0+

9. u(x) = 1 + ln x −

1

0+

ln(xt2 )u(t)dt, 0 < x 1

146

4 Fredholm Integral Equations

1

x u(t)dt, 0 < x 1 t2 π tu(t)dt 11. u(x) = sin x + (π − 1) cos x − cos x

10. u(x) = 1 + ln x −

ln

0+

0

π π 12. u(x) = x + sin x − cos x + x cos tu(t)dt 2 0 π π π 4 4 13. u(x) = 1 + sec2 x − sec2 xu(t)dt 14. u(x) = 1 − sec2 xu(t)dt 2 0 0 2 1 3 1 xtu(t)dt 16. u(x) = 2x + ex − xtu(t)dt 15. u(x) = 1 + x + ex − 3 0 4 0

4.2.6 The Successive Approximations Method The successive approximations method, or the Picard iteration method was introduced before in Chapter 3. The method provides a scheme that can be used for solving initial value problems or integral equations. This method solves any problem by ﬁnding successive approximations to the solution by starting with an initial guess as u0 (x), called the zeroth approximation. As will be seen, the zeroth approximation is any selective real-valued function that will be used in a recurrence relation to determine the other approximations. The most commonly used values for the zeroth approximations are 0, 1, or x. Of course, other real values can be selected as well. Given Fredholm integral equation of the second kind b K(x, t)u(t)dt, (4.163) u(x) = f (x) + λ a

where u(x) is the unknown function to be determined, K(x, t) is the kernel, and λ is a parameter. The successive approximations method introduces the recurrence relation u0 (x) = any selective real valued function, b (4.164) K(x, t)un (t)dt, n 0. un+1 (x) = f (x) + λ a

The question of convergence of un (x) is justiﬁed by Theorem 3.1. At the limit, the solution is determined by using the limit u(x) = lim un+1 (x). n→∞

(4.165)

It is interesting to point out that the Adomian decomposition method admits the use of an iteration formula of the form u0 (x) = all terms not included inside the integral sign, b u1 (x) = λ K(x, t)u0 (t)dt, a

4.2 Fredholm Integral Equations of the Second Kind

u2 (x) = λ

147

b

K(x, t)u1 (t)dt,

(4.166)

a

.. .

b

un+1 (x) = un (x) + λ

K(x, t)un (t)dt. a

The diﬀerence between the two formulas (4.164) and (4.166) can be summarized as follows: 1. The successive approximations method gives successive approximations of the solution u(x), whereas the Adomian method gives successive components of the solution u(x). 2. The successive approximations method admits the use of a selective real-valued function for the zeroth approximation u0 , whereas the Adomian decomposition method assigns all terms that are not inside the integral sign for the zeroth component u0 (x). Recall that this assignment was modiﬁed when using the modiﬁed decomposition method. 3. The successive approximations method gives the exact solution, if it exists, by u(x) = lim un+1 (x). (4.167) n→∞

However, the Adomian decomposition method gives the solution as inﬁnite series of components by ∞ u(x) = un (x). (4.168) n=0

This series solution converges rapidly to the exact solution if such a solution exists. The successive approximations method, or the iteration method will be illustrated by studying the following examples. Example 4.23 Solve the Fredholm integral equation by using the successive approximations method u(x) = x + ex −

1

xtu(t)dt.

(4.169)

0

For the zeroth approximation u0 (x), we can select u0 (x) = 0.

(4.170)

The method of successive approximations admits the use of the iteration formula 1 x un+1 (x) = x + e − xtun (t)dt, n 0. (4.171) 0

Substituting (4.170) into (4.171) we obtain

148

4 Fredholm Integral Equations

u1 (x) = x + ex − u2 (x) = x + ex − u3 (x) = x + ex − .. .

1 0 1 0 1 0

un+1 = x + ex −

1 0

xtu0 (t)dt = ex + x, 1 xtu1 (t)dt = ex − x, 3 1 xtu2 (t)dt = ex + x, 9

xtun (t)dt = ex +

(4.172)

(−1)n x. 3n

Consequently, the solution u(x) of (4.169) is given by u(x) = lim un+1 (x) = ex . n→∞

(4.173)

Example 4.24 Solve the Fredholm integral equation by using the successive approximations method 1

xtu(t)dt.

u(x) = x + λ

(4.174)

−1

For the zeroth approximation u0 (x), we can select (4.175) u0 (x) = x. The method of successive approximations admits the use of the iteration formula 1

un+1 (x) = x + λ

−1

xtun (t)dt, n 0.

(4.176)

Substituting (4.175) into (4.176) we obtain 2 u1 (x) = x + λx, 3 2 2 2 u2 (x) = x + λx + λ2 x, 3 3 3 2 2 2 2 u3 (x) = x + λx + λ2 x + λ3 x, (4.177) 3 3 3 .. . 2 3 n+1 2 2 2 2 2 3 un+1 (x) = x + λx + λ x+ λ x + ···+ λn+1 x. 3 3 3 3 The solution u(x) of (4.174) is given by 3 3x , 0<λ< , (4.178) u(x) = lim un+1 (x) = n→∞ 3 − 2λ 2

4.2 Fredholm Integral Equations of the Second Kind

149

obtained upon using the inﬁnite geometric series for the right side of (4.177). Example 4.25 Solve the Fredholm integral equation by using the successive approximations method π 2

u(x) = sin x + sin x

cos tu(t)dt.

(4.179)

0

For the zeroth approximation u0 (x), we select u0 (x) = 0. We next use the iteration formula un+1 (x) = sin x + sin x

π 2

0

(4.180)

cos tun (t)dt,

n 0.

(4.181)

Substituting (4.180) into (4.181) we obtain 3 sin x, 2 15 u4 (x) = sin x, 8

(4.182)

u(x) = lim un+1 (x) = 2 sin x.

(4.183)

u1 (x) = sin x,

u2 (x) =

7 u3 (x) = sin x, 4 .. . 2n+1 − 1 1 sin x = 2 − un+1(x) = sin x. 2n 2n The solution u(x) of (4.179) is given by n→∞

Example 4.26 Solve the Fredholm integral equation by using the successive approximations method π u(x) = x + sec2 x −

4

xu(t)dt.

(4.184)

0

For the zeroth approximation u0 (x), we may select u0 (x) = 0. We next use the iteration formula un+1(x) = x + sec2 x −

π 4

0

xun (t)dt,

(4.185) n 0.

(4.186)

This in turn gives u1 (x) = sec2 x + x, π4 u3 (x) = sec2 x + x, 1024 .. .

π2 x, 32 π6 u4 (x) = sec2 x − x, 32768 u2 (x) = sec2 x −

(4.187)

150

4 Fredholm Integral Equations

un+1 (x) = sec2 x + (−1)n Notice that

lim

n→∞

π2 32

π2 32

n x.

n = 0.

(4.188)

Consequently, the solution u(x) of (4.184) is given by u(x) = lim un+1 (x) = sec2 x.

(4.189)

n→∞

Exercises 4.2.6 Use the successive approximations method to solve the following Fredholm integral equations: 1 1 1. u(x) = x + λ xtu(t)dt 2. u(x) = 1 + x3 + λ xtu(t)dt −1

3. u(x) = ex + 2xe−1 − 5. u(x) = 2x + sec2 x −

−1

1 −1

π 4

−π

4. u(x) = 2 + 2x + ex −

xtu(t)dt

1 0

xtu(t)dt

xu(t)dt

4

π π 4 xu(t)dt 6. u(x) = 1 + 1 + x + sec2 x − 4 0 π π 2 xtu(t)dt 7. u(x) = − 1 x + cos x − 2 0 3 π π xtu(t)dt 8. u(x) = + 3 x − cos x − 3 0 π π π3 2 2 xtu(t)dt 10. u(x) = x + sin x − xtu(t)dt 9. u(x) = x − sin x − 24 0 0 π π 3 11. u(x) = 1 + x(1 + u(t))dt x + sec x tan x − 3 0 π 3 3 x sec tu(t)dt 12. u(x) = x + sec x tan x − 2 0 π √ 4 13. u(x) = 2 x + sec x + tan x − x sec tu(t)dt π 14. u(x) = x + sin x + cos x − 8

0

0

π 4

x sin tu(t)dt

15. u(x) = (π + 2)x + sin x − cos x − 16. u(x) = x + ln(xt) −

π

0

1 0+

x(3 + u(t))dt

x(1 + u(t))dt

4.2 Fredholm Integral Equations of the Second Kind

151

4.2.7 The Series Solution Method A real function u(x) is called analytic if it has derivatives of all orders such that the Taylor series at any point b in its domain k u(n) (b) (4.190) u(x) = (x − b)n , n! n=0 converges to f (x) in a neighborhood of b. For simplicity, the generic form of Taylor series at x = 0 can be written as ∞ u(x) = an xn . (4.191) n=0

Following the discussion presented before in Chapter 3, the series solution method that stems mainly from the Taylor series for analytic functions, will be used for solving Fredholm integral equations. We will assume that the solution u(x) of the Fredholm integral equations b K(x, t)u(t)dt (4.192) u(x) = f (x) + λ a

is analytic, and therefore possesses a Taylor series of the form given in (4.191), where the coeﬃcients an will be determined recurrently. Substituting (4.191) into both sides of (4.192) gives ∞ b ∞ n n dt, (4.193) an x = T (f (x)) + λ K(x, t) an t n=0

a

or for simplicity we use a0 + a1 x + a2 x2 + · · · = T (f (x)) + λ

n=0

b

K(x, t) a0 + a1 t + a2 t2 + · · · dt,

a

(4.194) where T (f (x)) is the Taylor series for f (x). The integral equation (4.192) will be converted to a traditional integral in (4.193) or (4.194) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integral in (4.193) or (4.194), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x in both sides of the resulting equation to obtain a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (4.191). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained

152

4 Fredholm Integral Equations

series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. It is worth noting that using the series solution method for solving Fredholm integral equations gives exact solutions if the solution u(x) is a polynomial. However, if the solution is any other elementary function such as sin x, ex , etc, the series method gives the exact solution after rounding few of the coeﬃcients aj , j 0. This will be illustrated by studying the following examples. Example 4.27 Solve the Fredholm integral equation by using the series solution method 1 u(x) = (x + 1)2 + (xt + x2 t2 )u(t)dt. (4.195) −1

Substituting u(x) by the series u(x) =

∞

a n xn

(4.196)

n=0

into both sides of Eq. (4.195) leads to 1 ∞ ∞ n 2 2 2 n an x = (x + 1) + (an t ) dt. (xt + x t ) −1

n=0

(4.197)

n=0

Evaluating the integral at the right side gives 2 2 2 2 a0 + a1 x + a2 x2 + a3 x3 + · · · = 1 + 2 + a1 + a3 + a5 + a7 x 3 5 7 9 2 2 2 2 2 + 1 + a0 + a2 + a4 + a6 + a8 x2 . 3 5 7 9 11 (4.198) Equating the coeﬃcients of like powers of x in both sides of (4.198) gives 25 , an = 0, n 3. a0 = 1, a1 = 6, a2 = (4.199) 9 The exact solution is given by 25 (4.200) u(x) = 1 + 6x + x2 , 9 obtained upon substituting (4.199) into (4.196). Example 4.28 Solve the Fredholm integral equation by using the series solution method 1 2 3 (1 + xt)u(t)dt. (4.201) u(x) = x − x + 0

Substituting u(x) by the series u(x) =

∞ n=0

an xn ,

(4.202)

4.2 Fredholm Integral Equations of the Second Kind

into both sides of Eq. (4.201) leads to 1 ∞ ∞ n 2 3 n an x = x − x + (an t ) dt. (1 + xt) 0

n=0

153

(4.203)

n=0

Evaluating the integral at the right side, and equating the coeﬃcients of like powers of x in both sides of the resulting equation we ﬁnd 29 1 a0 = − , a1 = − , a2 = 1, a3 = −1, an = 0, n 4. (4.204) 60 6 Consequently, the exact solution is given by 29 1 u(x) = − − x + x2 − x3 . (4.205) 60 6 Example 4.29 Solve the Fredholm integral equation by using the series solution method 1 (xt2 − x2 t)u(t)dt. (4.206) u(x) = −x4 + −1

Substituting u(x) by the series u(x) =

∞

an xn ,

(4.207)

n=0

into both sides of Eq. (4.206) leads to 1 ∞ ∞ n 4 2 2 n (xt − x t) an x = −x + an t dt. n=0

−1

(4.208)

n=0

Evaluating the integral at the right side, and equating the coeﬃcients of like powers of x in both sides of the resulting equation we ﬁnd 30 20 a0 = 0, a1 = − , a2 = , a3 = 0, a4 = −1 (4.209) 133 133 an = 0, n 5. Consequently, the exact solution is given by 20 2 30 x+ x − x4 . (4.210) u(x) = − 133 133 Example 4.30 Solve the Fredholm integral equation by using the series solution method π2 u(x) = −1 + cos x + u(t)dt. (4.211) 0

Substituting u(x) by the series u(x) =

∞ n=0

into both sides of Eq. (4.211) gives

an xn ,

(4.212)

154

4 Fredholm Integral Equations ∞

an xn = −1 + cos x +

n=0

π 2

0

∞

(an tn ) dt.

(4.213)

n=0

Evaluating the integral at the right side, using the Taylor series of cos x, and proceeding as before we ﬁnd (−1)j , j 0. (4.214) a0 = 1, a2j+1 = 0, a2j = (2j)! Consequently, the exact solution is given by u(x) = cos x. (4.215)

Exercises 4.2.7 Use the series solution method to solve the following Fredholm integral equations: 1 1 (1 − 3xt)u(t)dt 2. u(x) = 6x + 4x2 + (xt2 − x2 t)u(t)dt 1. u(x) = 1 + 0

3. u(x) = 5x − 2x2 + 4. u(x) = 5x +

1

−1

−1

1

−1

(x2 t3 − x3 t2 )u(t)dt

(1 − xt)u(t)dt

5. u(x) = 2 + 5x − 3x2 + 6. u(x) = 3x − 5x3 +

1

−1

1

−1

(1 − xt)u(t)dt

(1 − xt)u(t)dt

7. u(x) = 2 − 2x + 5x4 + 7x5 + 8. u(x) = 3x2 − 5x3 +

1 −1

9. u(x) = −2 + sin x + 11. u(x) = sec2 x − 1 +

π 2

−π

1 −1

(x − t)u(t)dt

x4 tu(t)dt tu(t)dt

10. u(x) = −2 + x2 + sin x +

2

π 4

0

u(t)dt

12. u(x) = −1 + ln(1 + x) +

π 2

−π

tu(t)dt

2

e−1

0

u(t)dt

4.3 Homogeneous Fredholm Integral Equation Substituting f (x) = 0 into the Fredholm integral equation of the second kind b u(x) = f (x) + λ K(x, t)u(t)dt, (4.216) a

the homogeneous Fredholm integral equation of the second kind is given by

4.3 Homogeneous Fredholm Integral Equation

155

b

u(x) = λ

K(x, t)u(t)dt.

(4.217)

a

In this section we will focus our study on the homogeneous Fredholm integral equation (4.217) for separable kernel K(x, t) only. The main goal for studying the homogeneous Fredholm equation is to ﬁnd nontrivial solution, because the trivial solution u(x) = 0 is a solution of this equation. Moreover, the Adomian decomposition method is not applicable here because it depends mainly on assigning a non-zero value for the zeroth component u0 (x), and in this kind of equations f (x) = 0. Based on this, the direct computation method will be employed here to handle this kind of equations.

4.3.1 The Direct Computation Method The direct computation method was used before in this chapter. This method replaces the homogeneous Fredholm integral equations by a single algebraic equation or by a system of simultaneous algebraic equations depending on the number of terms of the separable kernel K(x, t). As stated before, the direct computation method handles Fredholm integral equations in a direct manner and gives the solution in an exact form but not in a series form as Adomian method or the successive approximations method. It is important to point out that this method will be applied for the degenerate or separable kernels of the form n gk (x)hk (t). (4.218) K(x, t) = k=1

The direct computation method can be applied as follows: 1. We ﬁrst substitute (4.218) into the homogeneous Fredholm integral equation the form: b u(x) = λ K(x, t) u(t)dt. (4.219) a

2. This substitution leads to b h1 (t)u(t)dt + λg2 (x) u(x) = λg1 (x) a

b

h2 (t)u(t)dt + · · · a

b

+λgn (x)

(4.220)

hn (t)u(t)dt. a

3. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (4.220) becomes u(x) = λα1 g1 (x) + λα2 g2 (x) + · · · + λαn gn (x), where

(4.221)

156

4 Fredholm Integral Equations

b

hi (t) u(t)dt, 1 i n.

αi =

(4.222)

a

4. Substituting (4.221) into (4.222) gives a system of n simultaneous algebraic equations that can be solved to determine the constants αi , 1 i n. Using the obtained numerical values of αi into (4.221), the solution u(x) of the homogeneous Fredholm integral equation (4.217) follows immediately. Example 4.31 Solve the homogeneous Fredholm integral equation by using the direct computation method π2 cos x sin t u(t)dt. (4.223) u(x) = λ 0

This equation can be rewritten as u(x) = αλ cos x, where

π 2

α=

sin t u(t)dt.

(4.224) (4.225)

0

Substituting (4.224) into (4.225) gives π2 cos t sin tdt, α = αλ

(4.226)

0

that gives 1 αλ. (4.227) 2 Recall that α = 0 gives the trivial solution. For α = 0, we ﬁnd that the eigenvalue λ is given by λ = 2. (4.228) α=

This in turn gives the eigenfunction u(x) by u(x) = A cos x,

(4.229)

where A is a non zero arbitrary constant, with A = 2α. Example 4.32 Solve the homogeneous Fredholm integral equation by using the direct computation method 1 2ex+t u(t)dt. (4.230) u(x) = λ 0

This equation can be rewritten as u(x) = 2αλex , where

α= 0

1

et u(t)dt.

(4.231) (4.232)

4.3 Homogeneous Fredholm Integral Equation

Substituting (4.231) into (4.232) gives α = 2αλ

1

157

e2t dt,

(4.233)

0

that gives

α = αλ(e2 − 1).

(4.234)

Recall that α = 0 gives the trivial solution. For α = 0, we ﬁnd that the eigenvalue λ is given by 1 . (4.235) λ= 2 e −1 This in turn gives the eigenfunction u(x) by A u(x) = 2 ex , e −1 where A is a non zero arbitrary constant, with A = 2α.

(4.236)

Example 4.33 Solve the homogeneous Fredholm integral equation by using the direct computation method π sin(x + t) u(t)dt. (4.237) u(x) = λ 0

Notice that the kernel sin(x+t) = sin x cos t+cos x sin t is separable. Equation (4.237) can be rewritten as u(x) = αλ sin x + βλ cos x,

where

π

cos tu(t)dt,

α= 0

β=

(4.238)

π

sin tu(t)dt.

(4.239)

0

Substituting (4.238) into (4.239) gives π α= cos t(αλ sin t + βλ cos t)dt, 0

(4.240)

π

sin t(αλ sin t + βλ cos t)dt,

β= 0

that gives 1 1 βλπ, β = αλπ. 2 2 For α = 0, β = 0, we ﬁnd that the eigenvalue λ is given by 2 λ = ± , α = β. π This in turn gives the eigenfunction u(x) by A u(x) = ± (sin x + cos x), π α=

(4.241)

(4.242)

(4.243)

158

4 Fredholm Integral Equations

where A = 2α. Example 4.34 Solve the homogeneous Fredholm integral equation by using the direct computation method 1 u(x) = λ (12x + 2t) u(t)dt. (4.244) −1

Equation (4.244) can be rewritten as u(x) = 12αλx + 2βλ, where

1

u(t)dt,

α=

1

β=

−1

(4.245)

tu(t)dt.

(4.246)

−1

Substituting (4.245) into (4.246) gives 1 (12αλt + 2βλ)dt = 4βλ, α= −1 1

β=

(4.247) t(12αλt + 2βλ)dt = 8αλ.

−1

Recall that α = 0 and β = 0 give the trivial solution. For α = 0, β = 0, we ﬁnd that the eigenvalue λ is given by √ 1 λ = ± √ , β = 2α. (4.248) 4 2 This in turn gives the eigenfunction u(x) by √ α (4.249) u(x) = ± √ (6x + 2). 2 2 Example 4.35 Solve the homogeneous Fredholm integral equation by using the direct computation method 1 u(x) = λ 10(x2 − 2xt − t2 ) u(t)dt. (4.250) 0

Equation (4.250) can be rewritten as u(x) = 10αλx2 − 20βλx − 10γλ, where

1

u(t)dt,

α= 0

β=

1

tu(t)dt, 0

γ=

1

t2 u(t)dt.

(4.251)

(4.252)

0

Substituting (4.251) into (4.252) gives 1 10 α= αλ − 10βλ − 10λγ, (10αλt2 − 20βλt − 10γλ)dt = 3 0 1 20 5 β= t(10αλt2 − 20βλt − 10γλ)dt = αλ − βλ − 5λγ, (4.253) 2 3 0

4.4 Fredholm Integral Equations of the First Kind

1

γ= 0

159

t2 (10αλt2 − 20βλt − 10γλ)dt = 2αλ − 5βλ −

10 λγ. 3

Recall that α = 0, β = 0 and γ = 0 give the trivial solution. For α = 0, β = 0 and γ = 0, and by solving the system of equations we ﬁnd 20 7 3 γ, β = γ, (4.254) λ=− , α= 5 3 3 and γ is left as a free parameter. This in turn gives the eigenfunction u(x) by (4.255) u(x) = γ(−40x2 + 28x + 6), where γ is a non-zero arbitrary constant.

Exercises 4.3 Use the direct computation method to solve the homogeneous Fredholm integral equations: π π 3 1. u(x) = λ sin2 xu(t)dt 2. u(x) = λ tan x sec tu(t)dt 3. u(x) = λ 5. u(x) = λ 7. u(x) = λ 9. u(x) = λ

0

π 4

10 sec2 xu(t)dt

4. u(x) = λ

xtu(t)dt

6. u(x) = λ

8 sin−1 xtu(t)dt

8. u(x) = λ

(x + t)u(t)dt

10. u(x) = λ

0

2 0 1 0

−1

0

π

sin xu(t)dt

0 1

π

1

0 1 −1

12. u(x) = λ

(2 − 3x − 3t)u(t)dt

14. u(x) = λ

8 cos −1 xtu(t)dt

0

1 cos(x − t)u(t)dt π

xet u(t)dt

0

1

11. u(x) = λ 13. u(x) = λ

1

−1 1 0 1

(x − 10t2 )u(t)dt

(3 − 6x + 9t)u(t)dt (−12x2 + 24xt + 18t2 )u(t)dt

0

4.4 Fredholm Integral Equations of the First Kind We close this chapter by studying Fredholm integral equations of the ﬁrst kind given by b f (x) = λ K(x, t)u(t)dt, x ∈ D, (4.256) a

where D is a closed and bounded set in real numbers, and f (x) is the data. The range of x does not necessarily coincide with the range of integration [7]. The unknown function u(x), that will be determined, occurs only inside the integral sign and this causes special diﬃculties. The kernel K(x, t) and the function f (x) are given real-valued functions, and λ is a parameter that

160

4 Fredholm Integral Equations

is often omitted. However, the parameter λ plays an important role in the singular cases and in the bifurcation points as will be seen later in the text. An important remark has been reported in [7] and other references concerning the data function f (x). The function f (x) must lie in the range of the kernel K(x, t) [7]. For example, if we set the kernel by K(x, t) = sin x sin t. (4.257) Then if we substitute any integrable function u(x) in (4.256), and we evaluate the integral, the resulting f (x) must clearly be a multiple of sin x [7]. This means that if f (x) is not a multiple of the x component of the kernel, then a solution for (4.256) does not exist. This necessary condition on f (x) can be generalized. In other words, the data function f (x) must contain components which are matched by the corresponding x components of the kernel K(x, t) Fredholm integral equation of the ﬁrst kind is considered ill-posed problem. Hadamard [8] postulated the following three properties: 1. Existence of a solution. 2. Uniqueness of a solution. 3. Continuous dependence of the solution u(x) on the data f (x). This property means that small errors in the data f (x) should cause small errors [9] in the solution u(x). A problem is called a well-posed problem if it satisﬁes the three aforementioned properties. Problems that are not well-posed are called ill-posed problems such as inverse problems. Inverse problems are ill-posed problems that might not have a solution in the true sense, if a solution exists it may not be unique, and the obtained solution might not depend continuously on the observed data. If the kernel K(x, t) is smooth, then the Fredholm integral equation (4.256) is very often ill-posed and the solution u(x) is very sensitive to any change in the data f (x). In other words, a very small change on the data f (x) can give a large change in the solution u(x). For all these reasons, the Fredholm integral equations of the ﬁrst kind is ill-posed that may have no solution, or if a solution exists it is not unique and may not depend continuously on the data. The Fredholm integral equations of the ﬁrst kind (4.256) appear in many physical models such as radiography, stereology, spectroscopy, cosmic radiation, image processing and electromagnetic ﬁelds. Fredholm integral equations of the ﬁrst kind arise naturally in the theory of signal processing. Many inverse problems in science and engineering lead to the Fredholm integral equations of the ﬁrst kind. An inverse problem is a process where the solution u(x) can be obtained by solving (4.256) from the observed data f (x) at various values of x. Most inverse problems are ill-posed problems. This means that the Fredholm integral equations of the ﬁrst kind is aill-posed problem, and solving this equation may lead to a lot of diﬃculties. Several methods have been used to handle the Fredholm integral equations of the ﬁrst kind. The Legendre wavelets, the augmented Galerkin method, and the collocation method are examples of the methods used to handle this

4.4 Fredholm Integral Equations of the First Kind

161

equation. The methods that we used so far in this text cannot handle this kind of equations independently if it is expressed in its standard form (4.256). However, in this text, we will ﬁrst apply the method of regularization that received a considerable amount of interest, especially in solving ﬁrst order integral equations. The method transforms ﬁrst kind equation to second kind equation. We will second apply the homotopy perturbation method [10] to handle speciﬁc cases of the Fredholm integral equations where the kernel K(x, t) is separable. In what follows we will present a brief summary of the method of regularization and the homotopy perturbation method that will be used to handle the Fredholm integral equations of the ﬁrst kind.

4.4.1 The Method of Regularization The method of regularization was established independently by Phillips [11] and Tikhonov [12]. The method of regularization consists of replacing illposed problem by well-posed problem. The method of regularization transforms the linear Fredholm integral equation of the ﬁrst kind b f (x) = K(x, t)u(t)dt, x ∈ D, (4.258) a

to the approximation Fredholm integral equation b K(x, t)uμ (t)dt, x ∈ D, μuμ (x) = f (x) −

(4.259)

a

where μ is a small positive parameter. It is clear that (4.259) is a Fredholm integral equation of the second kind that can be rewritten 1 b 1 uμ (x) = f (x) − K(x, t)uμ (t)dt, x ∈ D. (4.260) μ μ a Moreover, it was proved in [7,13] that the solution uμ of equation (4.260) converges to the solution u(x) of (4.258) as μ → 0 according to the following lemma [14]: Lemma 4.1 Suppose that the integral operator of (4.258) is continuous and coercive in the Hilbert space where f (x), u(x), and uμ (x) are deﬁned, then: 1. |uμ | is bounded independently of μ, and 2. |uμ (x) − u(x)| → 0 when μ → 0. The proof of this lemma can be found in [7,13]. In summary, by combining the method of regulariztion with any of the methods used before for solving Fredholm integral equation of the second

162

4 Fredholm Integral Equations

kind, we can solve Fredholm integral equation of the ﬁrst kind (4.258). The method of regulariztion transforms the ﬁrst kind to a second kind. The resulting integral equation (4.260) can be solved by any of the methods that were presented before in this chapter. The exact solution u(x) of (4.258) can thus be obtained by u(x) = lim uμ (x). (4.261) μ→0

In what follows we will present ﬁve illustrative examples where we will use the method of regulariztion to transform the ﬁrst kind equation to a second kind equation. The resulting equation will be solved by any appropriate method that we used before. Example 4.36 Combine the method of regulariztion and the direct computation method to solve the Fredholm integral equation of the ﬁrst kind 14 1 x e = ex−t u(t)dt. (4.262) 4 0 Using the method of regularization, Equation (4.262) can be transformed to 1 1 x 1 4 x−t e uμ (t)dt. (4.263) uμ (x) = e − 4μ μ 0 The resulting Fredholm integral equation of the second kind will be solved by the direct computation method. Equation (4.263) can be written as 1 α (4.264) uμ (x) = − ex , 4μ μ where 1 4

α= 0

e−t uμ (t)dt.

(4.265)

To determine α, we substitute (4.264) into (4.265), integrate the resulting integral and solve to ﬁnd that 1 α= . (4.266) 1 + 4μ This in turn gives ex uμ (x) = . (4.267) 1 + 4μ The exact solution u(x) of (4.262) can be obtained by u(x) = lim uμ (x) = ex . μ→0

(4.268)

Example 4.37 Combine the method of regulariztion and the direct computation method to solve the Fredholm integral equation of the ﬁrst kind

4.4 Fredholm Integral Equations of the First Kind

ex + 1 =

1

163

(4tex + 3) u(t)dt.

(4.269)

0

Notice that the data function f (x) = ex + 1 contains components which are matched by the corresponding x components of the kernel K(x, t) = 4tex + 3. This is a necessary condition to guarantee a solution. Using the method of regularization, Equation (4.269) can be transformed to 1 1 1 1 (4tex + 3)uμ (t)dt. (4.270) uμ (x) = ex + − μ μ μ 0 The resulting Fredholm integral equation of the second kind will be solved by the direct computation method. Equation (4.270) can be written as 1 1 4α x 3β uμ (x) = − e + − , (4.271) μ μ μ μ where 1

α= 0

1

tuμ (t)dt,

β= 0

uμ (t)dt.

(4.272)

To determine α and β, we substitute (4.271) into (4.272), integrate the resulting integrals and solve to ﬁnd that −2(e + 6 + μe) 3(e − 3 − μ) , β=− α= . (4.273) 2(6e − 18 − 7μ − μ2 ) 6e − 18 − 7μ − μ2 Substituting this result into (4.271) gives the approximate solution (1 + μ)ex + (7 − 3e + μ) . (4.274) uμ (x) = 6(3 − e) + (7μ + μ2 ) The exact solution u(x) of (4.269) can be obtained by 1 7 − 3e u(x) = lim uμ (x) = ex + . (4.275) μ→0 6(3 − e) 6(3 − e) It is interesting to point out that another solution to this equation is given by (4.276) u(x) = x2 . As stated before, the Fredholm integral equation of the ﬁrst kind is ill-posed problem. For ill-posed problems, the solution might not exist, and if it exists, the solution may not be unique. Example 4.38 Combine the method of regulariztion and the direct computation method to solve the Fredholm integral equation of the ﬁrst kind π π sin x = sin(x − t) u(t)dt. (4.277) 2 0 Notice that the data function f (x) = π2 sin x contains component which is matched by the corresponding x component of the kernel K(x, t). This is a necessary condition to guarantee a solution.

164

to

4 Fredholm Integral Equations

Using the method of regularization, Equation (4.277) can be transformed 1 π π uμ (x) = sin x − sin(x − t)uμ (t)dt, (4.278) 2μ μ 0

that can be written as

uμ (x) =

where

α π − 2μ μ

sin x +

π

α= 0

cos tuμ (t)dt,

β= 0

β cos x, μ

(4.279)

sin tuμ (t)dt.

(4.280)

π

To determine α and β, we substitute (4.279) into (4.280), integrate the resulting integrals and solve to ﬁnd that π2 μ π3 . (4.281) , β = α= 2(π 2 + 4μ2 ) π 2 + 4μ2 Substituting this result into (4.279) gives the approximate solution 2πμ π2 uμ (x) = 2 sin x + 2 cos x. (4.282) 2 π + 4μ π + 4μ2 The exact solution u(x) of (4.277) can be obtained by (4.283) u(x) = lim uμ (x) = cos x. μ→0

Example 4.39 Combine the method of regulariztion and the Adomian decomposition method to solve the Fredholm integral equation of the ﬁrst kind 13 1 −x e = et−x u(t)dt. (4.284) 3 0 Using the method of regularization, Equation (4.284) can be transformed to 1 1 −x 1 3 t−x e uμ (t)dt. (4.285) uμ (x) = e − 3μ μ 0 The Adomian decomposition method admits the use of ∞ uμ (x) = uμn (x), (4.286) n=0

and the recurrence relation 1 −x e , 3μ 1 1 3 t−x e uμk (t)dt, uμk+1 (x) = − μ 0 This in turn gives the components uμ0 (x) =

(4.287) k 0.

4.4 Fredholm Integral Equations of the First Kind

165

1 −x 1 e , uμ1 (x) = − 2 e−x , 3μ 9μ (4.288) 1 −x 1 −x e , u (x) = − e , uμ2 (x) = μ 3 27μ3 81μ2 and so on. Substituting this result into (4.286) gives the approximate solution 1 (4.289) uμ (x) = e−x . 1 + 3μ The exact solution u(x) of (4.284) can be obtained by uμ0 (x) =

u(x) = lim uμ (x) = e−x . μ→0

(4.290)

Example 4.40 Combine the method of regulariztion and the successive approximations method to solve the Fredholm integral equation of the ﬁrst kind 1 1 xt u(t)dt. (4.291) x= 4 0 Using the method of regularization, Equation (4.291) can be transformed to 1 1 1 xtuμ (t)dt. (4.292) uμ (x) = x− 4μ μ 0 To use the successive approximations method, we ﬁrst select uμ0 (x) = 0. Consequently, we obtain the following approximations uμ0 (x) = 0, 1 x, uμ1 (x) = 4μ 1 1 uμ2 (x) = x, x− (4.293) 4μ 12μ2 1 1 1 x+ x, uμ3 (x) = x− 4μ 12μ2 36μ3 1 1 1 1 x+ x− x, uμ4 (x) = x− 4μ 12μ2 36μ3 108μ4 and so on. Based on this we obtain the approximate solution 3 x. (4.294) uμ (x) = 4(1 + 3μ) The exact solution u(x) of (4.291) can be obtained by 3 u(x) = lim uμ (x) = x. (4.295) μ→0 4 It is interesting to point out that another solution to this equation is given by (4.296) u(x) = x2 . As stated before, the Fredholm integral equation of the ﬁrst kind is ill-posed problem and the solution may not be unique.

166

4 Fredholm Integral Equations

Exercises 4.4.1 Combine the regularization method with any other method to solve the Fredholm integral equations of the ﬁrst kind 1 1 1 1 2 1. (1 − e−2 )e3x = e3x−4tu(t)dt 2. e3x = e3x−3t u(t)dt 2 2 0 0 1 1 6 3 xt2 u(t)dt 4. x2 = x2 t2 u(t)dt 3. x = 4 5 0 0 1 1 2 1 5. x2 = x2 t2 u(t)dt 6. x = xtu(t)dt 5 5 −1 0 1 1 2 1 x2 t2 u(t)dt 8. x2 = x2 t2 u(t)dt 7. x2 = 6 3 0 −1 1 1 1 1 9. − x = xtu(t)dt 10. x = xtu(t)dt 4 4 0 0 1 1 1 7 11. x= x= xtu(t)dt 12. xtu(t)dt 12 12 0 0 π π π π 13. sin x = cos(x − t)u(t)dt 14. cos x = cos(x − t)u(t)dt 2 2 0 0 π π (x − t)u(t)dt 16. 2 + π − 2x = (x − t)u(t)dt 15. 2 − π + 2x = 0

0

4.4.2 The Homotopy Perturbation Method The homotopy perturbation method was introduced and developed by JiHuan He in [10] and was used recently in the literature for solving linear and nonlinear problems. The homotopy perturbation method couples a homotopy technique of topology and a perturbation technique. A homotopy with an embedding parameter p ∈ [0, 1] is constructed, and the impeding parameter p is considered a small parameter. The method was derived and illustrated in [10], and several diﬀerential equations were examined. The coupling of the perturbation method and the homotopy method has eliminated the limitations of the traditional perturbation technique [10]. In what follows we illustrate the homotopy perturbation method to handle Fredholm integral equations of the second kind and the ﬁrst kind.

The HPM for Fredholm Integral Equation of the Second Kind In what follows we present the homotopy perturbation method for handling the Fredholm integral equations of the second kind. We ﬁrst consider the Fredholm integral equation of the second kind

4.4 Fredholm Integral Equations of the First Kind

167

b

v(x) = f (x) +

K(x, t)v(t)dt.

(4.297)

a

We now deﬁne the operator

b

L(u) = u(x) − f (x) −

K(x, t)u(t)dt = 0,

(4.298)

a

where u(x) = v(x). Next we deﬁne the homotopy H(u, p), p ∈ [0, 1] by H(u, 0) = F (u), H(u, 1) = L(u), (4.299) where F (u) is a functional operator. We construct a convex homotopy of the form H(u, p) = (1 − p)F (u) + pL(u) = 0. (4.300) This homotopy satisﬁes (4.299) for p = 0 and p = 1 respectively. The embedding parameter p monotonically increases from 0 to 1 as the trivial problem F (u) = 0 continuously deformed [10] to the original problem L(u) = 0. The homotopy perturbation method admits the use of the expansion ∞ u= pn un , (4.301) n=0

and consequently v = lim

p→1

∞

pn un .

(4.302)

n=0

The series (4.302) converges to the exact solution if such a solution exists. Substituting (4.301) into (4.300), using F (x) = u(x) − f (x), and equating the terms with like powers of the embedding parameter p we obtain the recurrence relation b 0 n+1 p : u0 (x) = f (x), p : un+1 = K(x, t)un (t)dt, n 0. (4.303) a

Notice that the recurrence relation (4.303) is the same standard Adomian decomposition method as presented before in this chapter. This proves the following theorem: Theorem 4.3 The Adomian decomposition method is a homotopy perturbation method with a convex homotopy given by b K(x, t)un (t)dt = 0. (4.304) H(u, p) = u(x) − f (x) − p a

HPM for Fredholm Integral Equation of the First Kind In what follows we present the homotopy perturbation method for handling the Fredholm integral equations of the ﬁrst kind of the form b f (x) = K(x, t)v(t)dt. (4.305) a

168

4 Fredholm Integral Equations

We now deﬁne the operator

L(u) = f (x) −

b

K(x, t)u(t)dt = 0.

(4.306)

a

We construct a convex homotopy of the form H(u, p) = (1 − p)u(x) + pL(u)(x) = 0. (4.307) The embedding parameter p monotonically increases from 0 to 1. The homotopy perturbation method admits the use of the expansion ∞ p n un , (4.308) u= n=0

and consequently v(x) = lim

p→1

∞

pn un (x).

(4.309)

n=0

The series (4.309) converges to the exact solution if such a solution exists. Substituting (4.308) into (4.307), and proceeding as before we obtain the recurrence relation u0 (x) = 0, u1 (x) = f (x), b (4.310) un+1 (x) = un (x) − K(x, t)un (t)dt, n 1. a

If the kernel is separable, i.e. K(x, t) = g(x)h(t), then the following condition b K(t, t)dt < 1, (4.311) 1 − a must be justiﬁed for convergence. The proof of this condition is left to the reader. We will concern ourselves only on the case where K(x, t) = g(x)h(t). The HPM will be used to solve the following Fredholm integral equations of the ﬁrst kind. Example 4.41 Use the homotopy perturbation method to solve the Fredholm integral equation of the ﬁrst kind 13 1 x e = ex−t u(t)dt. (4.312) 3 0 Notice that

13 2 K(t, t)dt = < 1. 1 − 3 0

Using the recurrence relation (4.310) we ﬁnd

(4.313)

4.4 Fredholm Integral Equations of the First Kind

u0 (x) = 0,

1 x e , 3

u1 (x) =

un+1 (x) = un (x) −

1 3

169

e

x−t

0

un (t)dt,

(4.314) n 1.

This in turn gives u0 (x) = 0,

u1 (x) =

u2 (x) = u1 (x) −

1 3

0

u3 (x) = u2 (x) −

1 3

0

1 3

1 x e , 3 ex−tu1 (t)dt = e

x−t

2 x e , 9

4 x e , u2 (t)dt = 27

8 x e , 81 0 and so on. Consequently, the approximate solution is given by 1 2 4 8 x u(x) = e + + + + ··· , 3 9 27 81 that converges to the exact solution u(x) = ex . u4 (x) = u3 (x) −

(4.315)

ex−tu3 (t)dt =

(4.316)

(4.317)

Example 4.42 Use the homotopy perturbation method to solve the Fredholm integral equation of the ﬁrst kind 14 1 −x e = et−x u(t)dt. (4.318) 4 0 14 3 K(t, t)dt = < 1. 1 − 4 0

Notice that

Using the recurrence relation (4.310) we ﬁnd 1 u0 (x) = 0, u1 (x) = e−x , 4 14 un+1 (x) = un (x) − et−x un (t)dt, 0

(4.319)

(4.320) n 1.

This in turn gives u0 (x) = 0,

u1 (x) =

u2 (x) = u1 (x) −

0

u3 (x) = u2 (x) −

1 4

0

1 4

1 −x e , 4 et−x u1 (t)dt =

3 −x e , 16

et−x u2 (t)dt =

9 −x e , 64

(4.321)

170

4 Fredholm Integral Equations

u4 (x) = u3 (x) −

1 4

0

et−x u3 (t)dt =

27 −x e , 256

and so on. Consequently, the approximate solution is given by 1 3 9 27 u(x) = e−x + + + + ··· , 4 16 64 256 that converges to the exact solution u(x) = e−x ,

(4.322)

(4.323)

obtained by evaluating the sum of the inﬁnite geometric series. Example 4.43 Use the homotopy perturbation method to solve the Fredholm integral equation of the ﬁrst kind 1

xt u(t)dt.

(4.324)

2 K(t, t)dt = < 1. 3

(4.325)

x= 0

Notice that

1 −

0

1

Using the recurrence relation (4.310) we ﬁnd u0 (x) = 0, u1 (x) = x, 1 un+1(x) = un (x) − xtun (t)dt, 0

n 1.

(4.326)

This in turn gives u0 (x) = 0,

u1 (x) = x, 1 2 xtu1 (t)dt = x, u2 (x) = u1 (x) − 3 0 1 4 xtu2 (t)dt = x, u3 (x) = u2 (x) − 9 0 1 8 xtu3 (t)dt = u4 (x) = u3 (x) − x, 27 0 and so on. Consequently, the approximate solution is given by 8 2 4 + ··· , u(x) = x 1 + + + 3 9 27 that converges to the exact solution u(x) = 3x.

(4.327)

(4.328)

(4.329)

Example 4.44 Use the homotopy perturbation method to solve the Fredholm integral equation of the ﬁrst kind 1 5 x= xt u(t)dt. (4.330) 6 0

4.4 Fredholm Integral Equations of the First Kind

Notice that

1 −

1

0

171

2 K(t, t)dt = < 1. 3

Using the recurrence relation (4.310) we ﬁnd 5 u0 (x) = 0, u1 (x) = x, 6 1 un+1(x) = un (x) − xtun (t)dt, 0

(4.331)

(4.332) n 1.

This in turn gives u0 (x) = 0,

u1 (x) =

u2 (x) = u1 (x) − u3 (x) = u2 (x) −

1

xtu1 (t)dt =

0

1

0

5 x, 6 5 x, 9

10 x, xtu2 (t)dt = 27

(4.333)

1

20 x, 81 0 and so on. Consequently, the approximate solution is given by 5 2 4 8 5 + ··· , (4.334) u(x) = x + x 1 + + + 6 9 3 9 27 that converges to the exact solution 5 u(x) = x, (4.335) 2 obtained by evaluating the sum of the inﬁnite geometric series. It is interesting to point out that another solution to this equation is given by u(x) = 1 + x. (4.336) u4 (x) = u3 (x) −

xtu3 (t)dt =

As stated before, the Fredholm integral equation of the ﬁrst kind is ill-posed problem. For ill-posed problems, the solution might not exist, and if it exists, the solution may not be unique. Example 4.45 Use the homotopy perturbation method to solve the Fredholm integral equation of the ﬁrst kind 1 1 x= xt u(t)dt. (4.337) 4 0 Notice that

1 −

0

1

2 K(t, t)dt = < 1. 3

Using the recurrence relation (4.310) we ﬁnd

(4.338)

172

4 Fredholm Integral Equations

1 u1 (x) = x, 4 1 un+1(x) = un (x) − xtun (t)dt, u0 (x) = 0,

0

(4.339) n 1.

This in turn gives 1 u1 (x) = x, 4 1 1 u2 (x) = u1 (x) − xtu1 (t)dt = x, 6 0 1 1 xtu2 (t)dt = x, u3 (x) = u2 (x) − 9 0 1 2 u4 (x) = u3 (x) − x, xtu3 (t)dt = 27 0 and so on. Consequently, the approximate solution is given by 1 2 4 8 u(x) = x 1 + + + + ··· , 4 3 9 27 that converges to the exact solution 3 u(x) = x. 4 It is interesting to point out that another solution to this equation by u(x) = x2 . u0 (x) = 0,

(4.340)

(4.341)

(4.342) is given (4.343)

As stated before, the Fredholm integral equation of the ﬁrst kind is ill-posed problem and the solution may not be unique. Notice that the ill-posed Fredholm problem is linear.

Exercises 4.4.2 Use the homotopy perturbation method to solve the Fredholm integral equations of the ﬁrst kind 1 1 1 1 2 1. (1 − e−2 )e3x = e3x−4tu(t)dt 2. e3x = e3x−3tu(t)dt 2 2 0 0 1 1 6 3 xt2 u(t)dt 4. x2 = x2 t2 u(t)dt 3. x = 4 5 0 0 1 1 1 2 x2 t2 u(t)dt 6. x = xtu(t)dt 5. x2 = 5 5 −1 0 1 1 1 2 7. x2 = x2 t2 u(t)dt 8. x2 = x2 t2 u(t)dt 6 3 0 −1 1 1 1 1 9. − x = xtu(t)dt 10. x = xtu(t)dt 4 4 0 0

References 11.

1 x= 12

173

1

0

xtu(t)dt

12.

7 x= 12

1 0

xtu(t)dt

References 1. W.V. Lovitt, Linear Integral Equations, Dover, New York, (1950). 2. G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994). 3. G. Adomian, A review of the decomposition method and some recent results for nonlinear equation, Math. Comput. Modelling, 13(1992) 17–43. 4. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 5. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc Singapore, (1997). 6. A.M.Wazwaz, Necessary conditions for the appearance of noise terms in decomposition solution series, Appl. Math. Comput., 81 (1997) 199–204. 7. L.M. Delves, and J. Walsh, Numerical Solution of Integral Equations, Oxford University Press, London, (1974). 8. J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Diﬀerential equations, Yale University Press, New Haven, (1923). 9. R. Kress, Linear Integral Equations, Springer, Berlin, (1999). 10. J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg., 178 (1999) 257–262. 11. D.L. Phillips, A technique for the numerical solution of certain integral equations of the ﬁrst kind, J. Assoc. Comput. Mach, 9 (1962) 84–96. 12. A.N. Tikhonov, On the solution of incorrectly posed problem and the method of regularization, Soviet Math, 4 (1963) 1035–1038. 13. R.F. Churchhouse, Handbook of Applicable Mathematics, Wiley, New York, (1981). 14. Y. Cherruault and V. Seng, The resolution of non-linear integral equations of the ﬁrst kind using the decomposition method of Adomian, Kybernetes, 26 (1997) 109–206.

Chapter 5

Volterra Integro-Diﬀerential Equations

5.1 Introduction Volterra studied the hereditary inﬂuences when he was examining a population growth model. The research work resulted in a speciﬁc topic, where both diﬀerential and integral operators appeared together in the same equation. This new type of equations was termed as Volterra integro-diﬀerential equations [1–4], given in the form x u(n) (x) = f (x) + λ K(x, t)u(t)dt, (5.1) n

0

where u(n) (x) = ddxnu . Because the resulted equation in (5.1) combines the diﬀerential operator and the integral operator, then it is necessary to deﬁne initial conditions u(0), u (0), . . . , u(n−1) (0) for the determination of the particular solution u(x) of the Volterra integro-diﬀerential equation (5.1). Any Volterra integro-diﬀerential equation is characterized by the existence of one or more of the derivatives u (x), u (x),. . . outside the integral sign. The Volterra integro-diﬀerential equations may be observed when we convert an initial value problem to an integral equation by using Leibnitz rule. The Volterra integro-diﬀerential equation appeared after its establishment by Volterra. It then appeared in many physical applications such as glassforming process, nanohydrodynamics, heat transfer, diﬀusion process in general, neutron diﬀusion and biological species coexisting together with increasing and decreasing rates of generating, and wind ripple in the desert. More details about the sources where these equations arise can be found in physics, biology and engineering applications books. To determine a solution for the integro-diﬀerential equation, the initial conditions should be given, and this may be clearly seen as a result of involving u(x) and its derivatives. The initial conditions are needed to determine the exact solution.

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

176

5 Volterra Integro-Diﬀerential Equations

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind In what follows we will present the recently developed methods, namely the Adomian decomposition method (ADM) and the variational iteration method (VIM) that will be used to handle the Volterra integro-diﬀerential equations of the second kind. Moreover, some of the traditional methods, namely the Laplace transform method, the series solution method, converting Volterra integro-diﬀerential equations to equivalent Volterra integral equations and converting Volterra integro-diﬀerential equations to equivalent initial value problem, will be studied as well. However, the Volterra integro-diﬀerential equations of the ﬁrst kind will be examined. The Laplace transform method and the variational iteration method will be used to handle the ﬁrst kind equations.

5.2.1 The Adomian Decomposition Method The Adomian decomposition method [5–9] gives the solution in an inﬁnite series of components that can be recurrently determined. The obtained series may give the exact solution if such a solution exists. Otherwise, the series gives an approximation for the solution that gives high accuracy level. Without loss of generality, we may assume a Volterra integro-diﬀerential equation of the second kind given by x u (x) = f (x) + K(x, t)u(t)dt, u(0) = a0 , u (0) = a1 . (5.2) 0

Integrating both sides of (5.2) from 0 to x twice leads to x K(x, t)u(t)dt , u(x) = a0 + a1 x + L−1 (f (x)) + L−1

(5.3)

0

where the initial conditions u(0) and u (0) are used, and L−1 is a two-fold integral operator. We then use the decomposition series ∞ un (x), (5.4) u(x) = n=0

into both sides of (5.3) to obtain ∞ −1 −1 un (x) = a0 + a1 x + L (f (x)) + L n=0

0

x

K(x, t)

∞

un (t) dt ,

n=0

(5.5) or equivalently

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

u0 (x) + u1 (x) + u2 (x) + u3 (x) + · · · = a0 + a1 x + L−1 (f (x)) x x K(x, t)u0 (t)dt + L−1 K(x, t)u1 (t)dt +L−1 0 0 x +L−1 K(x, t)u2 (t)dt + · · · .

177

(5.6)

0

To determine the components u0 (x), u1 (x), u2 (x), u3 (x), . . . of the solution u(x), we set the recurrence relation u0 (x) = a0 + a1 x + L−1 (f (x)) , x (5.7) uk+1 (x) = L−1 K(x, t)uk (t)dt , k 0, 0

where the zeroth component u0 (x) is deﬁned by all terms not included inside the integral sign of (5.6). Having determined the components ui (x), i 0, the solution u(x) of (5.2) is then obtained in a series form. Using (5.4), the obtained series converges to the exact solution if such a solution exists. n However, for concrete problems, a truncated series k=0 uk (x) is usually used to approximate the solution u(x) that can be used for numerical purposes. Remarks 1. The Adomian decomposition method was presented before to handle a second order Volterra integro-diﬀerential equations. For other orders, we can follow the same approach as presented. This will be explained later in details by discussing the illustrative examples, where ﬁrst-order, second-order, thirdorder, and fourth-order Volterra integro-diﬀerential equations will be studied. 2. The modiﬁed decomposition method that we used before can be used for handling Volterra integro-diﬀerential equations of any order. 3. The phenomenon of the noise terms that was applied before can be used here if noise terms appear. The Adomian decomposition method for solving the second kind Volterra integro-diﬀerential equations will be illustrated by studying the following examples. The selected equations are of orders 1, 2, 3, and 4. Other equations of higher orders can be treated in a like manner. Example 5.1 Use the Adomian method to solve the Volterra integro-diﬀerential equation x u(t)dt, u(0) = 0. (5.8) u (x) = 1 − 0

Applying the integral operator L−1 deﬁned by x −1 L (·) = (·)dx,

(5.9)

0

to both sides of (5.8), i.e. integrating both sides of (5.8) once from 0 to x, and using the given initial condition we obtain

178

5 Volterra Integro-Diﬀerential Equations

u(x) = x − L

−1

x

u(t)dt .

(5.10)

0

Using the decomposition series (5.4), and using the recurrence relation (5.7) we obtain u0 (x) = x, x 1 −1 u1 (x) = −L u0 (t)dt = − x3 , 3! 0 x (5.11) 1 u1 (t)dt = x5 , u2 (x) = L−1 5! 0 x 1 u3 (x) = L−1 u2 (t)dt = − x7 , 7! 0 and so on. This gives the solution in a series form 1 1 1 u(x) = x − x3 + x5 − x7 + · · · , 3! 5! 7! and hence the exact solution is given by u(x) = sin x.

(5.12) (5.13)

Example 5.2 Use the Adomian method to solve the Volterra integro-diﬀerential equation x u (x) = 1 + x + (x − t)u(t)dt, u(0) = 1, u (0) = 1. (5.14) 0

Applying the two-fold integral operator L−1 deﬁned by x x −1 L (·) = (·)dxdx, 0

(5.15)

0

to both sides of (5.14), i.e. integrating both sides of (5.14) twice from 0 to x, and using the given initial conditions we obtain x 1 3 1 2 −1 (x − t)u(t)dt . (5.16) u(x) = 1 + x + x + x + L 2! 3! 0 Using the decomposition series (5.4), and using the recurrence relation (5.7) we obtain 1 1 u0 (x) = 1 + x + x2 + x3 , 2! 3! x (5.17) (x − t)u0 (t)dt u1 (x) = L−1 0

1 1 1 1 4 x + x5 + x6 + x7 , 4! 5! 6! 7! and so on. This gives the solution in a series form 1 1 1 1 1 1 u(x) = 1 + x + x2 + x3 + x4 + x5 + x6 + x7 + · · · , 2! 3! 4! 5! 6! 7! and this converges to the exact solution =

(5.18)

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

u(x) = ex .

179

(5.19)

Example 5.3 Use the Adomian method to solve the Volterra integro-diﬀerential equation x u (x) = −1 + x − (x − t)u(t)dt, u(0) = 1, u (0) = −1, u (0) = 1. 0

Applying the three-fold integral operator L−1 deﬁned by x x x −1 (·)dxdxdx, L (·) = 0

0

(5.20) (5.21)

0

to both sides of (5.20), and using the given initial conditions we obtain x 1 1 1 (x − t)u(t)dt . (5.22) u(x) = 1 − x + x2 − x3 + x4 − L−1 2! 3! 4! 0 Using the decomposition series (5.4), and using the recurrence relation (5.7) we obtain 1 1 1 u0 (x) = 1 − x + x2 − x3 + x4 , 2! 3! 4! x −1 (5.23) (x − t)u0 (t)dt u1 (x) = −L 0

1 1 1 1 1 = − x5 + x6 − x7 + x8 − x9 , 5! 6! 7! 8! 9! and so on. This gives the solution in a series form 1 1 1 1 1 1 u(x) = 1 − x + x2 − x3 + x4 − x5 + x6 − x7 + · · · , 2! 3! 4! 5! 6! 7! and this converges to the exact solution u(x) = e−x .

(5.24) (5.25)

Example 5.4 Use the Adomian method to solve the Volterra integro-diﬀerential equation x (iv) (x − t)u(t)dt, u (x) = −1 + x − (5.26) 0 u(0) = −1, u (0) = 1, u (0) = 1, u (0) = −1. Applying the four-fold integral operator L−1 to both sides of (5.26) and using the given initial condition we obtain x u(x) = x − 1 − L−1 (x − t)u(t)dt . (5.27) 0

Using the decomposition series (5.4), and using the recurrence relation (5.7) we obtain

180

5 Volterra Integro-Diﬀerential Equations

1 2 1 1 1 x − x3 − x4 + x5 , 2! 3! 4! 5! x (5.28) 1 6 1 7 1 8 −1 (x − t)u0 (t)dt = x − x + x + · · · , u1 (x) = L 6! 7! 8! 0 and so on. The series solution is given by 1 1 1 1 1 1 u(x) = x − x3 + x5 − x7 + · · · − 1 − x2 + x4 − x6 + · · · , 3! 5! 7! 2! 4! 6! (5.29) so that the exact solution is given by u0 (x) = −1 + x +

u(x) = sin x − cos x.

(5.30)

Exercises 5.2.1 Solve the following Volterra integro-diﬀerential equations by using the Adomian decomposition method: x u(t)dt, u(0) = 0 1. u (x) = 1 + 0

2. u (x) = 1 + x +

x 0

3. u (x) = 2 + 4x + 8 4. u (x) = 1 +

x

0

5. u (x) = −1 +

(x − t)u(t)dt, u(0) = 1

x

0

(x − t)u(t)dt, u(0) = 1

(x − t)u(t)dt, u(0) = 1, u (0) = 0

x 0

6. u (x) = 1 + x +

(x − t)u(t)dt, u(0) = 1, u (0) = 0

x

0

7. u (x) = −1 − x +

(x − t)u(t)dt, u(0) = 1, u (0) = 1 x

0

(x − t)u(t)dt, u(0) = 1, u (0) = 1

8. u (x) = 2 − 2x sin x − 9. u (x) = −x −

1 3 x + 6

10. u (x) = 1 + x −

x

0 x 0

(x − t)u(t)dt, u(0) = 0, u (0) = 0

(x − t)u(t)dt, u(0) = 0, u (0) = 2

1 3 x + 31

x

0

11. u (x) = 1 − x + 2 sin x −

12. u (x) = 1 + x +

1 3 x + 6

13. u (x) = 1 + x −

1 4 x + 12

(x − t)u(t)dt, u(0) = 1, u (0) = 2 x

0 x

0

(x − t)u(t)dt, u(0) = 1, u (0) = 0, u (0) = 1

x

0

(x − t)u(t)dt, u(0) = 1, u (0) = −1, u (0) = −1

(x − t)u(t)dt, u(0) = 1, u (0) = 1, u (0) = 3

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind 181 x 14. u(iv) (x) = 1 + x − (x − t)u(t)dt, u(0) = u (0) = 1, u (0) = u (0) = −1 0

1 5 x + 15. u(iv) (x) = 1 + x − 20

x

0

(x − t)u(t)dt, u(0) = u (0) = u (0) = 1,

u (0) = 7 16. u(iv) (x) = 1 + x −

1 2 1 x − x3 + 2! 3!

x 0

(x − t)u(t)dt,

u(0) = u (0) = 2, u (0) = u (0) = 1

5.2.2 The Variational Iteration Method In Chapter 3, the variational iteration method was used to handle Volterra integral equations by converting it to an initial value problem or by converting it to an equivalent integro-diﬀerential equation. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the so called Adomian polynomials that we need for nonlinear problems. The standard ith order integro-diﬀerential equation is of the form x (i) u (x) = f (x) + K(x, t)u(t)dt, (5.31) 0 (i)

di u dxi ,

and u(0), u (0), . . . , u(i−1) (0) are the initial conditions. where u (x) = The correction functional for the integro-diﬀerential equation (5.31) is x ξ (i) un+1 (x) = un (x) + λ(ξ) un (ξ) − f (ξ) − K(ξ, r)˜ un (r) dr dξ. 0

0

(5.32) The variational iteration method [8,10] is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ that can be identiﬁed optimally via integration by parts and by using a restricted variation. Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), n 0 of the solution u(x). The zeroth approximation u0 (x) can be any selective function. However, the initial values u(0), u (0), · · · are preferably used for the selective zeroth approximation u0 (x) as will be seen later. Consequently, the solution is given by u(x) = lim un (x). n→∞

(5.33)

It is useful to summarize the Lagrange multipliers as derived in Chapter 3:

182

5 Volterra Integro-Diﬀerential Equations

u + f (u(ξ), u (ξ)) = 0, λ = −1, u + f (u(ξ), u (ξ), u (ξ)) = 0, λ = ξ − x, 1 (ξ − x)2 , 2! u(n) + f (u(ξ), u (ξ), u (ξ), · · · , u(n) (ξ)) = 0,

u + f (u(ξ), u (ξ), u (ξ), u (ξ)) = 0, λ = −

(5.34)

1 (ξ − x)(n−1) . (n − 1)! The VIM will be illustrated by studying the following examples. λ = (−1)n

Example 5.5 Use the variational iteration method to solve the Volterra integro-diﬀerential equation x

u (x) = 1 +

u(t)dt, u(0) = 1.

(5.35)

0

The correction functional for this equation is given by x ξ un+1 (x) = un (x) − un (ξ) − 1 − un (r)dr dξ, 0

(5.36)

0

where we used λ = −1 for ﬁrst-order integro-diﬀerential equations as shown in (5.34). We can use the initial condition to select u0 (x) = u(0) = 1. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1, x ξ u0 (ξ) − 1 − u1 (x) = u0 (x) − u0 (r)dr dξ 0

0

1 = 1 + x + x2 , 2! x u2 (x) = u1 (x) − u1 (ξ) − 1 − 0

0

ξ

u1 (r)dr

dξ

(5.37)

1 2 1 1 x + x3 + x4 , 2! 3! 4! x ξ u2 (r)dr dξ u3 (x) = u2 (x) − u2 (ξ) − 1 − = 1+x+

0

0

1 1 1 1 1 = 1 + x + x2 + x3 + x4 + x5 + x6 , 2! 3! 4! 5! 6! and so on. The VIM admits the use of u(x) = lim un(x),

(5.38)

u(x) = ex .

(5.39)

n→∞

that gives the exact solution

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

183

Example 5.6 Use the variational iteration method to solve the Volterra integro-diﬀerential equation x

u (x) = 1 +

0

(x − t)u(t)dt, u(0) = 1, u (0) = 0.

(5.40)

The correction functional for this equation is given by x ξ (ξ − x) un (ξ) − 1 − (ξ − r)un (r)dr dξ, (5.41) un+1 (x) = un (x) + 0

0

where λ = ξ − x for second-order integro-diﬀerential equations as shown in (5.34). We can use the initial conditions to select u0 (x) = u(0)+xu (0) = 1. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1, x ξ u1 (x) = u0 (x) + (ξ − x) u0 (ξ) − 1 − (ξ − r)u0 (r)dr dξ 0

0

1 1 = 1 + x2 + x4 , 2! 4! x ξ (ξ − x) u1 (ξ) − 1 − (ξ − r)u1 (r)dr dξ u2 (x) = u1 (x) + 0

0

(5.42)

1 2 1 1 1 x + x4 + x6 + x8 , 2! 4! 6! 8! x ξ (ξ − x) u2 (ξ) − 1 − (ξ − r)u2 (r)dr dξ u3 (x) = u2 (x) + = 1+

0

0

1 1 1 1 1 10 1 12 x + x , = 1 + x2 + x4 + x6 + x8 + 2! 4! 6! 8! 10! 12! and so on. The VIM admits the use of u(x) = lim un(x), n→∞

(5.43)

that gives the exact solution u(x) = cosh x.

(5.44)

Example 5.7 Use the variational iteration method to solve the Volterra integro-diﬀerential equation x 1 u (x) = 1 + x + x3 + (x − t)u(t)dt, u(0) = 1, u (0) = 0, u (0) = 1. 3! 0 (5.45) The correction functional for this equation is given by

184

5 Volterra Integro-Diﬀerential Equations

un+1 (x) = un(x) ξ 1 x 1 3 2 − (ξ − x) un (ξ) − 1 − ξ − ξ − (ξ − r)un (r)dr dξ, 2 0 3! 0 (5.46) where λ = − 12 (ξ − x)2 for third-order ODEs and for second-order integrodiﬀerential equations as shown in (5.34). The zeroth approximation u0 (x) can be selected by using the initial con1 2 1 2 ditions, hence we set u0 (x) = u(0) + xu (0) + 2! x u (0) = 1 + 2! x . Using this selection into the correction functional gives the following successive approximations 1 u0 (x) = 1 + x2 , 2! u1 (x) = u0 (x) ξ 1 x 1 ξ3 − − (ξ − x)2 u (ξ − r)u0 (r)dr dξ 0 (ξ) − 1 − ξ − 2 0 3! 0 1 1 1 1 1 1 = 1 + x2 + x3 + x4 + x5 + x6 + x7 , 2! 3! 4! 5! 6! 7! u2 (x) = u1 (x) ξ 1 x 1 3 2 − (ξ − x) u1 (ξ) − 1 − ξ − ξ − (ξ − r)u1 (r)dr dξ 2 0 3! 0 1 1 1 1 1 1 1 = 1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + · · · , 2! 3! 4! 5! 6! 7! 8! (5.47) and so on. The VIM admits the use of u(x) = lim un(x),

(5.48)

u(x) = ex − x.

(5.49)

n→∞

that gives the exact solution

Example 5.8 Use the variational iteration method to solve the Volterra integro-diﬀerential equation x 1 (x − t)u(t)dt, u(iv) (x) = 24 + x + x6 − 30 (5.50) 0 u(0) = 0, u (0) = 1, u (0) = 0, u (0) = −1. The correction functional for this equation is given by un+1 (x) = un (x) ξ 1 1 x ξ6 + (ξ − x)3 u(iv) (ξ − r)un (r)dr dξ, + n (ξ) − 24 − ξ − 6 0 30 0 (5.51)

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

185

where λ = 16 (ξ − x)3 for fourth-order ODEs integro-diﬀerential equations as shown in (5.34). 1 3 Using u0 (x) = x − 3! x into the correction functional gives the following successive approximations 1 u0 (x) = x − x3 , 3! u1 (x) = u0 (x) ξ 1 x 1 6 (iv) 3 + (ξ − x) u0 (ξ) − 24 − ξ − ξ + (ξ − r)u0 (r)dr dξ 6 0 30 0 1 1 1 1 1 x10 , = x4 + x − x3 + x5 − x7 + x9 + 3! 5! 7! 9! 151200 u2 (x) = u1 (x) ξ 1 x 1 (iv) + (ξ − x)3 u1 (ξ) − 24 − ξ − ξ 6 + (ξ − r)u1 (r)dr dξ 6 0 30 0 1 1 1 1 1 11 x + ··· , = x4 + x − x3 + x5 − x7 + x9 − 3! 5! 7! 9! 11! (5.52) and so on. Notice that x10 in u1 (x) has vanished from u2 (x). Similarly other noise terms vanish in the limit. The VIM admits the use of u(x) = lim un(x), n→∞

that gives the exact solution by u(x) = x4 + sin x.

(5.53) (5.54)

Exercises 5.2.2 Solve the following Volterra integro-diﬀerential equations by using the variational iteration method: x 1. u (x) = x − 1 − (x − t)u(t)dt, u(0) = 1 2. u (x) = 1 + x +

0 x 0

(x − t)u(t)dt, u(0) = 1

3. u (x) = 3ex − 2 − x + 4. u (x) = x + 5. u (x) = x −

x

0

x 0

(x − t)u(t)dt, u(0) = 0

(x − t)u(t)dt, u(0) = 0, u (0) = 1

1 3 x + 3!

x

0 x

(x − t)u(t)dt, u(0) = 0, u (0) = 2

1 3 x + (x − t)u(t)dt, u(0) = 1, u (0) = 1 3! 0 x (x − t)u(t)dt, u(0) = 1, u (0) = 1 7. u (x) = 1 + x + 6. u (x) = 1 −

0

186

5 Volterra Integro-Diﬀerential Equations

8. u (x) = −1 − x + 9. u (x) = −1 −

x 0

1 3 x + 3!

(x − t)u(t)dt, u(0) = 1, u (0) = 1

x

0

1 4 x + 12

10. u (x) = 2 − x −

(x − t)u(t)dt, u(0) = 1, u (0) = 1

11. u (x) = 1 − x + 2 sin x −

x

0

(x − t)u(t)dt, u(0) = 0, u (0) = 1 x

0

(x − t)u(t)dt, u(0) = 1, u (0) = −1, u (0) = −1

1 2 1 1 x (x − t)2 u(t)dt, x − x4 + 2! 4! 2 0 u(0) = 1, u (0) = 2, u (0) = 1 1 1 x 13. u (x) = −3 − 2x − x2 + 6ex + (x − t)2 u(t)dt, 2 2 0 u(0) = 0, u (0) = 1, u (0) = 2 x 14. u(iv) (x) = 1 + x − (x − t)u(t)dt, u(0) = u (0) = 1, u (0) = u (0) = −1

12. u (x) = 1 + x +

0

1 1 5 1 x x + 15. u (x) = 1 + x + x2 − (x − t)2 u(t)dt, u(0) = u (0) = u (0) = 1, 2! 60 2 0 u (0) = 3 1 1 1 x 16. u(iv) (x) = 1 + x + x2 − x3 + (x − t)2 u(t)dt, 2! 3 2 0 u(0) = 3, u (0) = u (0) = u (0) = 1 (iv)

5.2.3 The Laplace Transform Method The Laplace transform method was used before for solving Volterra integral equations of the ﬁrst and the second kind in Chapter 3. The details and properties of the Laplace transform method can be found in ordinary diﬀerential equations texts. Before we start applying this method, we summarize some of the concepts presented in Section 1.5. In the Laplace transform convolution theorem, it was stated that if the kernel K(x, t) of the integral equation x u(n) (x) = f (x) + λ K(x, t)u(t)dt, (5.55) 0

depends on the diﬀerence x − t, then it is called a diﬀerence kernel. The integro-diﬀerential equation can thus be expressed as x u(n) (x) = f (x) + λ K(x − t)u(t)dt. (5.56) 0

Consider two functions f1 (x) and f2 (x) that possess the conditions needed for the existence of Laplace transform for each. Let the Laplace transforms for the functions f1 (x) and f2 (x) be given by L{f1 (x)} = F1 (s),

L{f2 (x)} = F2 (s).

(5.57)

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

187

The Laplace convolution product of these two functions is deﬁned by x (f1 ∗ f2 )(x) = f1 (x − t)f2 (t)dt, (5.58)

or (f2 ∗ f1 )(x) =

0

x 0

f2 (x − t)f1 (t)dt.

(5.59)

Recall that (f1 ∗ f2 )(x) = (f2 ∗ f1 )(x).

(5.60)

We can easily show that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x L{(f1 ∗ f2 )(x)} = L f1 (x − t)f2 (t)dt = F1 (s)F2 (s). (5.61) 0

To solve Volterra integro-diﬀerential equations by using the Laplace transform method, it is essential to use the Laplace transforms of the derivatives of u(x). We can easily show that L{u(n) (x)} = sn L{u(x)} − sn−1 u(0) − sn−2 u (0) − · · · − u(n−1) (0). (5.62) This simply gives L{u (x)} = sL{u(x)} − u(0) = sU (s) − u(0), L{u (x)} = s2 L{u(x)} − su(0) − u (0) = s2 U (s) − su(0) − u (0), L{u (x)} = s3 L{u(x)} − s2 u(0) − su (0) − u (0) = s3 U (s) − s2 u(0) − su (0) − u (0),

(5.63)

L{u(iv) (x)} = s4 L{u(x)} − s3 u(0) − s2 u (0) − su (0) − u (0) = s4 U (s) − s3 u(0) − s2 u (0) − su (0) − u (0), and so on for derivatives of higher order. The Laplace transform method can be applied in a similar manner to the approach used before in Chapter 3. We ﬁrst apply the Laplace transform to both sides of (5.56), use the proper Laplace transform for the derivative of u(x), and then solve for U (s). We next use the inverse Laplace transform of both sides of the resulting equation to obtain the solution u(x) of the equation. Recall that Table 1.2 in Chapter 1 should be used. The Laplace transform method for solving Volterra integro-diﬀerential equations will be illustrated by studying the following examples. Example 5.9 Use the Laplace transform method to solve the Volterra integro-diﬀerential equation x u(t)dt, u(0) = 1. (5.64) u (x) = 1 + 0

188

5 Volterra Integro-Diﬀerential Equations

Notice that the kernel K(x − t) = 1. Taking Laplace transform of both sides of (5.64) gives L(u (x)) = L(1) + L(1 ∗ u(x)), (5.65) so that

1 1 + U (s), (5.66) s s obtained upon using (5.63). Using the given initial condition and solving for U (s) we ﬁnd 1 . (5.67) U (s) = s−1 sU (s) − u(0) =

By taking the inverse Laplace transform of both sides of (5.67), the exact solution is given by u(x) = ex . (5.68) Example 5.10 Use the Laplace transform method to solve the Volterra integro-diﬀerential equation x (x − t)u(t)dt, u(0) = 1, u (0) = 1. (5.69) u (x) = −1 − x + 0

Notice that the kernel K(x − t) = (x − t). Taking Laplace transform of both sides of (5.69) gives (5.70) L(u (x)) = −L(1) − L(x) + L((x − t) ∗ u(x)), so that

1 1 1 (5.71) s2 U (s) − su(0) − u (0) = − − 2 + 2 U (s), s s s obtained upon using (5.63). Using the given initial condition and solving for U (s) we ﬁnd 1 s U (s) = 2 + . (5.72) s + 1 s2 + 1 By taking the inverse Laplace transform of both sides of (5.72), the exact solution is given by u(x) = sin x + cos x. (5.73)

Example 5.11 Use the Laplace transform method to solve the Volterra integro-diﬀerential equation 1 2 1 4 1 x (x− t)2 u(t)dt, u(0) = 1, u (0) = 2, u = 1. u (x) = 1 + x+ x − x + 2! 4! 2 0 (5.74) Taking Laplace transform of both sides of (5.74) gives 1 1 1 L{u (x)} = L{1}+L{x}+ L{x2 }− L{x4 }+ L{(x−t)2 ∗u(x)}, (5.75) 2! 4! 2

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

189

so that

1 1 1 1 1 + + 3 − 5 + 3 U (s). s s2 s s s Using the given initial condition and solving for U (s) we ﬁnd 1 1 U (s) = 2 + . s s−1 Using the inverse Laplace transform, the exact solution is given by u(x) = x + ex . s3 U (s) − s2 − 2s − 1 =

(5.76)

(5.77) (5.78)

Example 5.12 Solve the Volterra integral equation by using by the Laplace transform method x sin(x − t)u(t)dt, u(iv) (x) = sin x + cos x + 2 (5.79) 0 u(0) = u (0) = u (0) = u (0) = 1. Taking Laplace transform of both sides of (5.79) gives L{u(iv) (x)} = L{sin x} + L{cos x} + 2L{sin(x − t) ∗ u(x)},

(5.80)

so that 1 s 2 + + U (s). s2 + 1 s2 + 1 s2 + 1 Using the given initial condition and solving for U (s) we ﬁnd 1 . U (s) = s−1 Using the inverse Laplace transform, the exact solution is given by u(x) = ex . s4 U (s) − s3 − s2 − s − 1 =

(5.81)

(5.82) (5.83)

Exercises 5.2.3 Solve the following Volterra integro-diﬀerential equations by using the Laplace transform method x 1. u (x) = 1 + x − x2 + (x − t)u(t)dt, u(0) = 3 0

1 2. u (x) = 2 + x − x3 + 3!

x 0

3. u (x) = −(a + b)u(x) − ab 4. u (x) = −x +

x

(x − t)u(t)dt, u(0) = 1

x 0

u(t)dt, u(0) = a − b

(x − t)u(t)dt, u(0) = 0, u (0) = 1 x 1 5. u (x) = −x − x3 + (x − t)u(t)dt, u(0) = 0, u (0) = 2 3! 0 x e−(x−t)u(t)dt, u(0) = 1, u (0) = 1 6. u (x) = cosh x + 0

0

190

5 Volterra Integro-Diﬀerential Equations

7. u (x) = cosh x − 8. u (x) = 4ex + 4

x 0 x

0

e(x−t) u(t)dt, u(0) = 1, u (0) = −1

e(x−t) u(t)dt, u(0) = 1, u (0) = 2

1 3 1 x x − (x − t)3 u(t)dt, u(0) = 0, u (0) = 1 3! 6 0 1 1 x (x − t)3 u(t)dt, u(0) = 1, u (0) = 0 10. u (x) = 1 + x2 + 2 6 0 x 11. u (x) = 1 + x − 2x2 + (x − t)u(t)dt, u(0) = 5, u (0) = u (0) = 1 9. u (x) = −x +

1 1 12. u (x) = 1 + x2 + 2! 2

0 x

0

(x − t)2 u(t)dt, u(0) = u (0) = 0, u (0) = 1

1 4 1 x x + (x − t)2 u(t)dt, u(0) = u (0) = u (0) = 1 4! 2 0 x 14. u(iv) (x) = e2x − e2(x−t) u(t)dt, u(0) = u (0) = u (0) = u (0) = 1 13. u (x) = x −

0

15. u(iv) (x) = 3ex + e2x −

x

0

e2(x−t) u(t)dt, u(0) = 0,

u (0) = 1, u (0) = 2, u (0) = 3 x 1 5 1 16. u(iv) (x) = − − x + e2x − e2(x−t) u(t)dt, 4 2 4 0 u(0) = 1, u (0) = 2, u (0) = u (0) = 1

5.2.4 The Series Solution Method It was stated before in Chapter 3 that a real function u(x) is called analytic if it has derivatives of all orders such that the Taylor series at any point b in its domain ∞ u(n) (b) (x − b)n , (5.84) u(x) = n! n=0 converges to u(x) in a neighborhood of b. For simplicity, the generic form of Taylor series at x = 0 can be written as ∞ a n xn . (5.85) u(x) = n=0

In this section we will apply the series solution method for solving Volterra integro-diﬀerential equations of the second kind. We will assume that the solution u(x) of the Volterra integro-diﬀerential equation x u(n) (x) = f (x)+λ K(x, t)u(t)dt, u(k) (0) = k!ak , 0 k (n−1), (5.86) 0

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

191

is analytic, and therefore possesses a Taylor series of the form given in (5.85), where the coeﬃcients an will be determined recurrently. The ﬁrst few coeﬃcients ak can be determined by using the initial conditions so that 1 1 a0 = u(0), a1 = u (0), a2 = u (0), a3 = u (0), (5.87) 2! 3! and so on. The remaining coeﬃcients ak of (5.85) will be determined by applying the series solution method to the Volterra integro-diﬀerential equation (5.86). Substituting (5.85) into both sides of (5.86) gives ∞ (n) ∞ x k k (5.88) ak x = T (f (x)) + K(x, t) ak t dt, 0

k=0

or for simplicity we use (a0 +a1 x+a2 x2 +· · · )(n) = T (f (x))+λ

k=0

x 0

K(x, t) a0 + a1 t + a2 t2 + · · · dt,

(5.89) where T (f (x)) is the Taylor series for f (x). The integro-diﬀerential equation (5.86) will be converted to a traditional integral in (5.88) or (5.89) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integral in (5.88) or (5.89), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0, where some of these coeﬃcients will be used from the initial conditions. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (5.85). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. Example 5.13 Use the series solution method to solve the Volterra integro-diﬀerential equation x u (x) = 1 +

u(t)dt.

(5.90)

0

Substituting u(x) by the series u(x) =

∞ n=0

into both sides of Eq. (5.90) leads to

an xn ,

(5.91)

192

5 Volterra Integro-Diﬀerential Equations

∞

n

an x

x

=1+ 0

n=0

∞

an t

n

dt.

(5.92)

n=0

Diﬀerentiating the left side once with respect to x, and evaluating the integral at the right side we ﬁnd ∞ ∞ 1 n−1 an−1 xn , nan x =1+ (5.93) n n=1 n=1 that can be rewritten as ∞ ∞ 1 n an−1 xn , (n + 1)an+1 x = 1 + a1 + n n=1 n=1

(5.94)

where we uniﬁed the exponent of x in both sides, and used a0 = 0 from the given initial condition. Equating the coeﬃcients of like powers of x in both sides of (5.94) gives the recurrence relation 1 an−1 , n 1. a0 = 0, a1 = 1, an+1 = (5.95) n(n + 1) where this result gives 1 a2n = 0, a2n+1 = , (5.96) (2n + 1)! for n 0. Substituting this result into (5.91) gives the series solution ∞ 1 (5.97) x2n+1 , u(x) = (2n + 1)! n=0 that converges to the exact solution u(x) = sinh x.

(5.98)

Example 5.14 Use the series solution method to solve the Volterra integro-diﬀerential equation x u (x) = 1 + x +

0

(x − t)u(t)dt,

u(0) = u (0) = 1.

(5.99)

Substituting u(x) by the series u(x) =

∞

an xn ,

(5.100)

n=0

into both sides of Eq. (5.99) leads to ∞ x ∞ n n (x − t) dt. an x =1+x+ an t n=0

0

(5.101)

n=0

Diﬀerentiating the left side twice, and by evaluating the integral at the right side we ﬁnd ∞ ∞ 1 an xn+2 , n(n − 1)an xn−2 = 1 + x + (5.102) (n + 2)(n + 1) n=2 n=0

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

or equivalently ∞ ∞ 2a2 +6a3 x+ (n+2)(n+1)an+2 xn = 1+x+ n=2

n=2

193

1 an−2 xn . (5.103) n(n − 1)

Using the initial conditions and equating the coeﬃcients of like powers of x in both sides of (5.103) gives the recurrence relation 1 1 a0 = 1, a1 = 1, a2 = , a3 = , 2! 3! (5.104) 1 an+2 = an−2 , n 2. (n + 2)(n + 1)n(n − 1) where this result gives 1 an = , (5.105) n! for n 0. Substituting this result into (5.100) gives the series solution ∞ 1 n x , (5.106) u(x) = n! n=0 that converges to the exact solution u(x) = ex .

(5.107)

Example 5.15 Use the series solution method to solve the Volterra integro-diﬀerential equation u (x) = 1 − x + 2 sin x −

x

0

(x − t)u(t)dt, u(0) = 1, u (0) = u (0) = −1.

(5.108)

Substituting u(x) by the series u(x) =

∞

an xn ,

(5.109)

n=0

into both sides of the equation (5.108), and using Taylor expansion for sin x we obtain ∞ x ∞ ∞ (−1)n 2n+1 n n an x =1−x+2 − an t (x − t) dt. x (2n + 1)! 0 n=0 n=0 n=0 (5.110) Diﬀerentiating the left side three times, and by evaluating the integral at the right side we ﬁnd 1 1 1 2 3 2 6a3 + 24a4x + 60a5 x + 120a6 x + · · · = 1 + x − a0 x − + a1 x3 + · · · , 2 3 6 (5.111) where we used few terms for simplicity reasons. Using the initial conditions and equating the coeﬃcients of like powers of x in both sides of (5.111) we ﬁnd

194

5 Volterra Integro-Diﬀerential Equations

a0 = 1,

a1 = −1,

1 , 2! 1 a5 = − . 5! a2 = −

1 1 , a4 = , 3! 4! Consequently, the series solution is given by ∞ ∞ (−1)n 2n (−1)n 2n+1 u(x) = x − x , (2n)! (2n + 1)! n=0 n=0 a3 =

that converges to the exact solution u(x) = cos x − sin x.

(5.112)

(5.113)

(5.114)

Example 5.16 Use the series solution method to solve the Volterra integro-diﬀerential equation u(iv) (x) = 1 + x −

x

0

(x − t)u(t)dt, u(0) = u (0) = 1, u (0) = u (0) = −1.

(5.115)

Proceeding as before we set (iv) ∞ n an x =1+x−

0

n=0

x

(x − t)

∞

an t

n

dt,

(5.116)

n=0

Diﬀerentiating the left side, integrating the right side, and using few terms for simplicity we ﬁnd 1 1 24a4 +120a5x+360a6x2 +840a7x3 +· · · = 1+x− a0 x2 − a1 x3 +· · · . (5.117) 2 6 Using the initial conditions and equating the coeﬃcients of like powers of x in both sides of (5.117) gives the recurrence relation 1 a0 = 1, a1 = 1, a2 = − , 2! (5.118) 1 1 1 1 a6 = − , · · · a3 = − , a 4 = , a 5 = , 3! 4! 5! 6! Consequently, the series solution is given by ∞ ∞ (−1)n 2n (−1)n 2n+1 , (5.119) u(x) = x + x (2n)! (2n + 1)! n=0 n=0 that converges to the exact solution u(x) = cos x + sin x.

(5.120)

Exercises 5.2.4 Solve the following Volterra integro-diﬀerential equations by using the series solution method: x 1. u (x) = 1 + u(t)dt, u(0) = 0 0

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind x 2. u (x) = 1 + x + (x − t)u(t)dt, u(0) = 1

195

0

x 1 3. u (x) = 2 − x + x3 − (x − t)u(t)dt, u(0) = −1 3! 0 x 4. u (x) = 1 + (x − t)u(t)dt, u(0) = 1, u (0) = 0 0

x 1 (x − t)u(t)dt, u(0) = 0, u (0) = 2 5. u (x) = x − x3 + 3! 0 x 6. u (x) = cosh x + e−(x−t)u(t)dt, u(0) = 1, u (0) = 1

0

7. u (x) = −1 − x +

x 0

(x − t)u(t)dt, u(0) = 1, u (0) = 1

x 1 4 x + (x − t)u(t)dt, u(0) = 0, u (0) = 1 12 0 1 1 x 9. u (x) = −x + x3 − (x − t)3 u(t)dt, u(0) = 0, u (0) = 1 3! 6 0 x 1 (x − t)u(t)dt, u(0) = 1, u (0) = 2 10. u (x) = 1 + x − x3 + 3! 0 x 1 11. u (x) = 1 + x + x3 + (x − t)u(t)dt, u(0) = 1, u (0) = 0, u (0) = 1 3! 0 1 1 x 12. u (x) = 1 + x2 + (x − t)2 u(t)dt, u(0) = u (0) = 0, u (0) = 1 2! 2 0 x 1 4 x + 13. u (x) = 1 + x − (x − t)u(t)dt, u(0) = 1, u (0) = 1, u (0) = 3 12 0 x 1 5 x + 14. u(iv) (x) = 1 + x − (x − t)u(t)dt, 20 0 u(0) = u (0) = u (0) = 1, u (0) = 7 x 15. u(iv) (x) = 3ex + e2x − e2(x−t) u(t)dt, u(0) = 0,

8. u (x) = 2 − x −

0

u (0) = 1, u (0) = 2, u (0) = 3

x 1 2 1 (x − t)u(t)dt, x − x3 + 2! 3! 0 u(0) = u (0) = 2, u (0) = u (0) = 1

16. u(iv) (x) = 1 + x −

5.2.5 Converting Volterra Integro-Diﬀerential Equations to Initial Value Problems The Volterra integro-diﬀerential equation x u(n) (x) = f (x)+λ K(x, t)u(t)dt, u(k) (0) = k!ak , 0 k (n−1), (5.121) 0

196

5 Volterra Integro-Diﬀerential Equations

can be solved by converting it to an equivalent initial value problem. The study will be concerned on the Volterra integro-diﬀerential equations where the kernel is a diﬀerence kernel of the form K(x, t) = K(x − t), such as x − t. The reason for this selection is that we seek ODEs with constant coeﬃcients. Having converted the integro-diﬀerential equation to an initial value problem, we then can use any standard method for solving ODEs. It is worth noting that the conversion process can be easily used, but it requires more works if compared with the integro-diﬀerential equations methods. The conversion process is obtained by diﬀerentiating both sides of the Volterra integro-diﬀerential equation as many times until we get rid of the integral sign. To perform the diﬀerentiation for the integral at the right side, the Leibnitz rule, that was introduced in Chapter 1, should be used. The initial conditions should be determined by using a variety of integral equations that we will obtain in the process of diﬀerentiation. To give a clear overview of this method we discuss the following illustrative examples. Example 5.17 Solve the Volterra integro-diﬀerential equation by converting it to an initial value problem x u (x) = 1 + u(t)dt, u(0) = 1. (5.122) 0

Diﬀerentiating both sides of (5.122) with respect to x and using the Leibnitz rule we ﬁnd u (x) = u(x), (5.123) with initial conditions given by (5.124) u(0) = 1, u (0) = 1. The second initial condition is obtained by substituting x = 0 in both sides of (5.122). The characteristic equation for the ODE (5.122) is r2 − 1 = 0, (5.125) which gives the roots r = ±1 so that the general solution is given by u(x) = Aex + Be−x .

(5.126) (5.127)

The constants A and B can be determined by using the initial conditions, where we ﬁnd A = 1, B = 0. The exact solution is (5.128) u(x) = ex . Example 5.18 Solve the Volterra integro-diﬀerential equation by converting it to an initial value problem x u (x) = 6 − 3x2 + u(t)dt, u(0) = 0. (5.129) 0

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

197

Diﬀerentiating both sides of (5.129) with respect to x and using the Leibnitz rule we ﬁnd u (x) − u(x) = −6x, (5.130) with initial conditions given by u(0) = 0, u (0) = 6. (5.131) The second initial condition is obtained by substituting x = 0 in both sides of (5.129). The characteristic equation for the related homogeneous ODE of (5.129) is r2 − 1 = 0, (5.132) which gives the roots r = ±1

(5.133)

so that the complementary solution is given by uc (x) = Aex + Be−x . To ﬁnd a particular solution up (x), we substitute up (x) = C + Dx,

(5.134) (5.135)

into (5.130) and equate coeﬃcients of like powers of x from both side to ﬁnd that C = 0, D = 6. This gives the general solution u(x) = uc (x) + up (x) = Aex + Be−x + 6x. (5.136) The constants A and B can be determined by using the initial conditions, where we ﬁnd A = 0, B = 0. The exact solution is given by u(x) = 6x. (5.137) Example 5.19 Solve the following Volterra integro-diﬀerential equation by converting it to an initial value problem x u (x) = 1 + x + (x − t)u(t)dt, u(0) = 1, u (0) = 1. (5.138) 0

Diﬀerentiating both sides and proceeding as before we ﬁnd x u(t)dt, u (x) = 1 +

(5.139)

0

and by diﬀerentiating again to remove the integral sign we ﬁnd u(iv) (x) = u(x).

(5.140) Two more initial conditions can be obtained by substituting x = 0 in (5.138) and (5.139), hence the four initial conditions are (5.141) u(0) = 1, u (0) = 1, u (0) = 1, u (0) = 1. The characteristic equation of (5.140) is r4 − 1 = 0, which gives the roots r = ±1, ±i, i =

√ −1,

(5.142) (5.143)

198

5 Volterra Integro-Diﬀerential Equations

so that the general solution is given by (5.144) u(x) = Aex + Be−x + C cos x + D sin x. The constants A, B, C, and D can be obtained by using the initial conditions where we found A = 1 and B = C = D = 0. The exact solution is given by u(x) = ex .

(5.145)

Example 5.20 Solve the following Volterra integro-diﬀerential equation by converting it to an initial value problem x (x − t)u(t)dt, u(0) = 1, u (0) = 1. (5.146) u (x) = −1 − x + 0

Diﬀerentiating both sides and proceeding as before we obtain x u (x) = −1 + u(t)dt,

(5.147)

0

and by diﬀerentiating again to remove the integral sign we ﬁnd u(iv) (x) = u(x).

(5.148)

Two more initial conditions can be obtained by substituting x = 0 in (5.146) and (5.147), hence the four initial conditions are u(0) = 1, u (0) = 1, u (0) = −1, The characteristic equation of (5.148) is

u (0) = −1.

r4 − 1 = 0, which gives the roots r = ±1, ±i, i =

√ −1,

(5.149) (5.150) (5.151)

so that the general solution is given by u(x) = Aex + Be−x + C cos x + D sin x. (5.152) The constants A, B, C, and D can be obtained by using the initial conditions where we found A = B = 0 and C = D = 1. The exact solution is given by u(x) = sin x + cos x.

(5.153)

Exercises 5.2.5 Solve the following Volterra integro-diﬀerential equations by converting the problem to an initial value problem: x 1. u (x) = 1 + u(t)dt, u(0) = 0 0

2. u (x) = −1 + x − 3. u (x) = 1 +

x 0

x

0

(x − t)u(t)dt, u(0) = 1

(x − t)u(t)dt, u(0) = 1, u (0) = 0

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind x 4. u (x) = x + (x − t)u(t)dt, u(0) = 0, u (0) = 1

199

0

x 1 3 x + (x − t)u(t)dt, u(0) = 1, u (0) = 1 3! 0 x 6. u (x) = −x + (x − t)u(t)dt, u(0) = 0, u (0) = 1 5. u (x) = −1 −

0

7. u (x) = 1 + x −

1 3 x + 3!

1 2 8. u (x) = − x2 − x3 + 2 3

x 0

x

0

(x − t)u(t)dt, u(0) = 1, u (0) = 2 (x − t)u(t)dt, u(0) = 1, u (0) = 4

5.2.6 Converting Volterra Integro-Diﬀerential Equation to Volterra Integral Equation The Volterra integro-diﬀerential equation x (n) u (x) = f (x) + λ K(x, t)u(t)dt,

(5.154)

0

can also be solved by converting it to an equivalent Volterra integral equation. Recall that in integro-diﬀerential equations, the initial conditions are usually prescribed. The study will be focused on the Volterra integro-diﬀerential equations where the kernel is a diﬀerence kernel. Having converted the integro-diﬀerential equation to an equivalent integral equation, the latter can be solved by using any of the methods presented in Chapter 3, such as Adomian method, series solution method, and Laplace transform method. It is obvious that the Volterra integro-diﬀerential equation (5.154) involves derivatives at the left side, and integral at the right side. To perform the conversion process, we need to integrate both sides n times to convert it to a standard Volterra integral equation. It is therefore useful to summarize some of formulas to support the conversion process. We point out that the ﬁrst set of formulas is usually studied in calculus. However, the second set is given in Section 1.4, where reducing multiple integrals to a single integral is presented. I. Integration of derivatives: from calculus we observe the following: x u (t)dt = u(x) − u(0), 0

x

0

x

0

x1

0 0 x1 x 2 0

u (t)dtdx1 = u(x) − xu (0) − u(0),

u (t)dtdx2 dx1 = u(x) −

and so on for other derivatives.

1 2 x u (0) − xu (0) − u(0), 2!

(5.155)

200

5 Volterra Integro-Diﬀerential Equations

II. Reducing multiple integrals to a single integral: from Chapter 1, we studied the following: x x1 x u(t)dtdx1 = (x − t)u(t)dt, 0 x

0 x1

0

0 x 0

0 x x1

0

x1

0

0

1 (x − t)u(t)dtdx1 = 2

1 (x − t) u(t)dtdx1 = 3 2

(x − t)3 u(t)dtdx1 =

1 4

x

0

0

(5.156)

x

3

(x − t) u(t)dt,

0

(x − t)2 u(t)dt,

x

(x − t)4 u(t)dt,

and so on. This can be generalized in the form xn−1 x x1 x 2 1 x ··· (x − t)u(t)dtdxn−1 · · · dx1 = (x − t)n u(t)dt. n! 0 0 0 0 0 (5.157) The conversion to an equivalent Volterra integral equation will be illustrated by studying the following examples. Example 5.21 Solve the following Volterra integro-diﬀerential equation by converting it to Volterra integral equation x u(t)dt, u(0) = 0. (5.158) u (x) = 1 + 0

Integrating both sides from 0 to x, and using the aforementioned formulas we ﬁnd x x u(x) − u(0) = x + u(t)dt. (5.159) 0

0

Using the initial condition gives the Volterra integral equation x (x − t)u(t)dt. u(x) = x +

(5.160)

0

We can select any of the proposed methods. The Adomian decomposition method will be used to solve this problem. Using the series assumption ∞ u(x) = un (x), (5.161) n=0

into both sides of the Volterra integral equation gives the recursion relation 1 1 1 (5.162) u0 (x) = x, u1 (x) = x3 , u2 (x) = x5 , u3 (x) = x7 , 3! 5! 7! so that the series solution is given by 1 1 1 (5.163) u(x) = x + x3 + x5 + x7 + · · · , 3! 5! 7! that converges to the exact solution

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

u(x) = sinh x.

201

(5.164)

Example 5.22 Solve the following Volterra integro-diﬀerential equation by converting it to Volterra integral equation x (x − t)u(t)dt, u(0) = 3. (5.165) u (x) = 1 + x − x2 + 0

Integrating both sides once from 0 to x and using the formulas given above we obtain 1 2 1 3 1 x u(x) − u(0) = x + x − x + (x − t)2 u(t)dt. (5.166) 2 3 2! 0 Notice that when we integrate the integral at the right side, we should use the formula (5.157) for the resulting double integral. By using the initial condition we obtain the Volterra integral equation 1 x 1 2 1 3 (x − t)2 u(t)dt. (5.167) u(x) = 3 + x + x − x + 2 3 2! 0 We proceed as before and use the Adomian decomposition method to ﬁnd 1 1 u0 (x) = 3 + x + x2 − x3 , 2 3 1 1 1 1 6 (5.168) u1 (x) = x3 + x4 + x5 − x , 2! 4! 5! 360 1 6 1 1 x + x7 + x8 + · · · . u2 (x) = 240 7! 8! Consequently, the series solution is given by 1 3 1 4 1 5 1 2 (5.169) u(x) = 2 + 1 + x + x + x + x + x + · · · , 2! 3! 4! 5! that converges to the exact solution u(x) = 2 + ex . (5.170) Example 5.23 Solve the following Volterra integro-diﬀerential equation by converting it to Volterra integral equation x (x − t)u(t)dt, u(0) = 0, u (0) = 1. (5.171) u (x) = −x + 0

Integrating both sides twice from 0 to x and using the formulas given above we ﬁnd 1 3 1 x (x − t)3 u(t)dt. (5.172) u(x) − xu (0) − u(0) = − x + 3! 3! 0 Notice that when we integrate the integral at the right side, we should use the formula (5.157) for the resulting triple integral. By using the initial conditions we obtain the Volterra integral equation

202

5 Volterra Integro-Diﬀerential Equations

1 3 1 x u(x) = x − x + (x − t)3 u(t)dt. (5.173) 3! 3! 0 We now select the successive approximations method and set the recurrence relation 1 1 x un (x) = x − x3 + (x − t)3 un−1 (t)dt, n 1. (5.174) 3! 3! 0 Selecting the zeroth approximation u0 (x) = x gives the approximations u0 (x) = x, 1 3 1 x + x5 , 3! 5! 1 1 1 1 u2 (x) = x − x3 + x5 − x7 + x9 . 3! 5! 7! 9! Consequently, the series solution is given by 1 1 1 1 u(x) = x − x3 + x5 − x7 + x9 + · · · , 3! 5! 7! 9! that converges to the exact solution u(x) = sin x. u1 (x) = x −

(5.175)

(5.176) (5.177)

Example 5.24 Solve the following Volterra integro-diﬀerential equation by converting it to Volterra integral equation x u (x) = 1 − x + 2 sin x − (x − t)u(t)dt, u(0) = 1, u (0) = −1, u (0) = −1. 0

(5.178) Integrating both sides three times from 0 to x and using the formulas given above we ﬁnd 1 3 1 4 1 x 1 2 2 (x − t)4 u(t)dt. u(x) + x + x − 1 = x − x + x + 2 cos x − 2 + 2! 3! 4! 4! 0 (5.179) By using the initial conditions we obtain the Volterra integral equation 1 1 1 x 1 (x − t)4 u(t)dt. (5.180) u(x) = −1 − x + x2 + x3 − x4 + 2 cos x + 2! 3! 4! 4! 0 Using the Adomian decomposition method gives the recurrence relation 1 1 1 u0 (x) = −1 − x + x2 + x3 − x4 + 2 cos x, 2! 3! 4! 1 3 1 5 1 6 (5.181) u1 (x) = 2x − x + x + x − 2 sin x + · · · , 3 5! 6! 1 1 6 x − 2 cos x + · · · , u2 (x) = 2 − x2 + x4 − 12 360 and so on. Consequently, the series solution is given by

5.3 Volterra Integro-Diﬀerential Equations of the First Kind

203

1 2 1 4 1 3 1 5 u(x) = 1 − x + x + · · · + x − x + x − · · · − 2 sin x, 2! 4! 3! 5! (5.182) that converges to the exact solution u(x) = cos x − sin x. (5.183)

Exercises 5.2.6 Solve the following Volterra integro-diﬀerential equations by converting the problem to Volterra integral equation: x x u(t)dt, u(0) = 0 2. u (x) = 1 − u(t)dt, u(0) = 1 1. u (x) = 1 − 0

3. u (x) = x +

0

x 0

4. u (x) = 1 + x + 5. u (x) = −1 −

(x − t)u(t)dt, u(0) = 0, u (0) = 1

x 0

(x − t)u(t)dt, u(0) = u (0) = 1

1 3 x + 3!

6. u (x) = 1 + x − 7. u (x) = 1 + x +

x

0

1 3 x + 3!

x 0

1 3 x + 3!

8. u (x) = 1 − e−x +

(x − t)u(t)dt, u(0) = 0, u (0) = 2

(x − t)u(t)dt, u(0) = 1, u (0) = 2

x

0

1 3 x + 3!

(x − t)u(t)dt, u(0) = u (0) = 1, u (0) = 0

x 0

(x − t)u(t)dt, u(0) = u (0) = 1, u (0) = 0

5.3 Volterra Integro-Diﬀerential Equations of the First Kind The standard form of the Volterra integro-diﬀerential equation [11–12] of the ﬁrst kind is given by x x K1 (x, t)u(t)dt + K2 (x, t)u(n) (t)dt = f (x), K2 (x, t) = 0, (5.184) 0

0

where initial conditions are prescribed. The Volterra integro-diﬀerential equation of the ﬁrst kind (5.184) can be converted into a Volterra integral equation of the second kind, for n = 1, by integrating the second integral in (5.184) by parts. The Volterra integro-diﬀerential equations of the ﬁrst kind will be handled in this section by Laplace transform method and the variational iteration method.

204

5 Volterra Integro-Diﬀerential Equations

5.3.1 Laplace Transform Method The Laplace transform method was used before for solving Volterra integral equations of the ﬁrst and the second kind in Chapter 3. It was also used in this chapter for solving integro-diﬀerential equations of the second kind. The analysis will be focused on equations where the kernels K1 (x, t) and K2 (x, t) of (5.184) are diﬀerence kernels. This means that each kernel depends on the diﬀerence (x − t). Taking Laplace transform of both sides of (5.184) gives (5.185) L(K1 (x − t) ∗ u(x)) + L(K2 (x − t) ∗ u(n) (x)) = L(f (x)), so that K1 (s)U (s)+K2 (s) sn U (s) − sn−1 u(0) − sn−2 u (0) − · · · − u(n−1) (0) = F (s), (5.186) where U (s) = L(u(x)), K1 (s) = L(K1 (x)),

K2 (s) = L(K2 (x)),

F (s) = L(f (x)). (5.187) Using the given initial conditions and solving for U (s) we ﬁnd

F (s) + K2 (s) sn−1 u(0) + sn−2 u (0) + · · · − u(n−1) (0) , (5.188) U (s) = K1 (s) + sn K2 (s) provided that (5.189) K1 (s) + sn K2 (s) = 0. By taking the inverse Laplace transform of both sides of (5.188), the exact solution is readily obtained. The analysis presented above can be explained by using the following illustrative examples. Example 5.25 Solve the following Volterra integro-diﬀerential equation of the ﬁrst kind x x (x − t)u(t)dt + (x − t)2 u (t)dt = 3x − 3 sin x, u(0) = 0. (5.190) 0

0

Taking Laplace transforms of both sides gives 1 2 3 3 U (s) + 3 (sU (s) − u(0)) = 2 − , (5.191) 2 s s s 1 + s2 where by using the given initial condition and solving for U (s) we obtain 1 U (s) = . (5.192) 1 + s2 Taking the inverse Laplace transform of both sides we ﬁnd u(x) = sin x. (5.193) Example 5.26 Solve the following Volterra integro-diﬀerential equation of the ﬁrst kind

5.3 Volterra Integro-Diﬀerential Equations of the First Kind

0

x

(x − t + 1)u (t)dt = 2xex + ex − x − 1,

u(0) = 0,

205

u (0) = 1. (5.194)

Notice in this equation that K1 (x, t) = 0, and K2 (x, t) = (x − t + 1). Taking Laplace transforms of both sides gives 1 2 1 1 1 1 + + (s2 U (s) − su(0) − u (0)) = − 2 − , (5.195) 2 2 s s (s − 1) s−1 s s where by using the given initial conditions and solving for U (s) we obtain 1 . (5.196) U (s) = (s − 1)2 Taking the inverse Laplace transform of both sides we ﬁnd u(x) = xex . (5.197) Example 5.27 Solve the following Volterra integro-diﬀerential equation of the ﬁrst kind x x cos(x − t)u(t)dt + sin(x − t)u (t)dt = 1 + sin x − cos x, (5.198) 0 0 u(0) = 1, u (0) = 1, u (0) = −1. Notice in this equation that K1 (x, t) = cos(x − t), and K2 (x, t) = sin(x − t). Taking Laplace transforms of both sides gives 1 1 s 1 s U (s) + (s3 U (s) − s2 u(0) − su(0) − u (0)) = + − , 2 2 2 1+s 1+s s 1+s 1 + s2 (5.199) where by using the given initial conditions and solving for U (s) we obtain 1 s U (s) = 2 + . (5.200) s 1 + s2 Taking the inverse Laplace transform of both sides we ﬁnd u(x) = x + cos x.

(5.201)

Example 5.28 Solve the following Volterra integro-diﬀerential equation of the ﬁrst kind x 1 x 1 (x − t)u(t)dt + (x − t − 1)u (t)dt = sin 2x, u(0) = 1, u (0) = 0. 4 0 2 0 (5.202) Taking Laplace transforms of both sides gives

1 1 1 2 1 1 s U (s) − su(0) − u (0) = 2 , (5.203) U (s) + − 2 2 s 4 s s s +4 where by using the given initial conditions and solving for U (s) we obtain s U (s) = . (5.204) 4 + s2 Taking the inverse Laplace transform of both sides we ﬁnd u(x) = cos 2x. (5.205)

206

5 Volterra Integro-Diﬀerential Equations

Exercises 5.3.1 Solve the following Volterra integro-diﬀerential equations of the ﬁrst kind by using Laplace transform method: x x 1 1. (x − t)u(t)dt + (x − t)2 u (t)dt = 3x + x2 − 3 sin x, u(0) = 0 2 0 0 x x (x − t)u(t)dt + (x − t)2 u (t)dt = 6 − 6 cos x − 3x sin x, u(0) = 0 2.

0 x

3. 0 x

4. 0 x

5. 0 x

6.

0 x

7. 0 x

8. 0

0

1 (x − t)u(t)dt − 2

x 0

(x − t + 1)u (t)dt = 2x2 −

(x − t + 1)u (t)dt = ex +

(x − t)u(t)dt +

x

0

sin(x − t)u(t)dt −

(x − t + 1)u (t)dt = 1 2

cosh(x − t)u(t)dt +

0

x

10. 0 x

11. 0 x

12. 0

1 2 x − 1, u(0) = 1 2

(x − t + 1)u (t)dt = 2 sin x − x, u(0) = −1, u (0) = 1

x

0 x

0

u(0) = 0, u (0) = 1 x cosh(x − t)u(t)dt + 9.

1 x, u(0) = 5 2

(x − t)u(t)dt +

1 2

(x − t)u(t)dt −

1 24

(x − t)2 u(t)dt −

x

0 x

x 0

sinh(x − t)u (t)dt =

1 x 1 xe − sinh x, 2 2

sinh(x − t)u (t)dt = xex , u(0) = u (0) = u (0) = 1 1 2 x , u(0) = u (0) = 1, u (0) = −1 2

(x − t)4 u (t)dt =

x

0

1 1 x − x cos x, u(0) = 0, u (0) = 1 2 2

(x − t)u (t)dt =

(x − t)2 u (t)dt =

0

1 12

1 3 x + sin x, u(0) = −1, u (0) = 1 6

1 3 x , u(0) = u (0) = 0, u (0) = 2, 3

(x − t)3 u (t)dt =

1 4 x , u(0) = u (0) = 0, u (0) = 3 12

5.3.2 The Variational Iteration Method The standard form of the Volterra integro-diﬀerential equation [11–12] of the ﬁrst kind is given by x x K1 (x, t)u(t)dt + K2 (x, t)u(n) (t)dt = f (x), K2 (x, t) = 0, (5.206) 0

0

where initial conditions are prescribed. We ﬁrst diﬀerentiate both sides of (5.206) to convert it to an equivalent Volterra integro-diﬀerential equation of the second kind, where Leibnitz rule should be used for diﬀerentiating the integrals at the left side, hence we obtain

5.3 Volterra Integro-Diﬀerential Equations of the First Kind

207

x ∂(K1 (x, t)) (n) K1 (x, x) 1 f (x) − u(x) − u (t)dt K2 (x, x) K2 (x, x) K2 (x, x) 0 ∂x x ∂(K2 (x, t)) (n) 1 u (t)dt, K2 (x, x) = 0. − K2 (x, x) 0 ∂x (5.207) As stated before, to use the variational iteration method, we should ﬁrst determine the Lagrange multiplier λ. The Lagrange multiplier λ can be determined based on the resulting integro-diﬀerential equations, where the following rule for λ 1 u(n) + f (u(t), u (t), u (t), · · · , u(n) (t)) = 0, λ = (−1)n (t − x)(n−1) , (n − 1)! (5.208) was derived. Having λ determined, an iteration formula, should be used for the determination of the successive approximations un+1 (x), n 0 of the solution u(x). The zeroth approximation u0 can be any selective function. However, using the initial values u(0), u (0), . . . are preferably used for the selective zeroth approximation u0 as will be seen later. Consequently, the solution is given by (5.209) u(x) = lim un(x). u(n) (x) =

n→∞

The VIM will be illustrated by studying the following examples. Example 5.29 Solve the Volterra integro-diﬀerential equation of the ﬁrst kind x 1 (x − t + 1)u (t)dt = ex + x2 − 1, u(0) = 1. (5.210) 2 0 Diﬀerentiating both sides of (5.210) once with respect to x gives the Volterra integro-diﬀerential equation of the second kind x u (x) = ex + x − u (t)dt. (5.211) 0

The correction functional for Eq. (5.211) is given by x t un (r)dr dt. un (t) − et − t + un+1 (x) = un (x) − 0

(5.212)

0

where we used λ(t) = −1. The zeroth approximation u0 (x) can be selected by u0 (x) = 1. This gives the successive approximations u0 (x) = 1, 1 2 x , 2! 1 1 u2 (x) = 1 + x + x2 − x3 , 2! 3! 1 1 u3 (x) = ex − x3 + x4 , 3! 4! u1 (x) = ex +

(5.213)

208

5 Volterra Integro-Diﬀerential Equations

1 2 x + 2! 1 u5 (x) = 1 + x + x2 + 2! .. . .

u4 (x) = 1 + x +

This gives

1 4 x − 4! 1 4 x + 4!

1 5 x , 5! 1 6 x 6!

1 1 1 un (x) = x + 1 + x2 + x4 + x6 + · · · . 2! 4! 6!

(5.214)

The exact solution is therefore given by u(x) = x + cosh x.

(5.215)

Example 5.30 Use the variational iteration method to solve the Volterra integro-diﬀerential equation of the ﬁrst kind x 1 x 1 (x − t)u(t)dt − (x − t + 1)u (t)dt = 2x2 − x, u(0) = 5. (5.216) 2 2 0 0 Diﬀerentiating both sides of (5.216) once with respect to x gives the Volterra integro-diﬀerential equation of the second kind x (2u(t) − u (t))dt. (5.217) u (x) = −8x + 1 + 0

The correction functional for equation this equation is given by x t un (t) + 8t − 1 − un+1 (x) = un (x) − (2un (r) − un (r)) dr dt. 0

0

(5.218) where we used λ(t) = −1. The zeroth approximation u0 (x) can be selected by u0 (x) = 5. This gives the successive approximations u0 (x) = 5, u1 (x) = 5 + x + x2 , 1 1 u2 (x) = 5 + x + x2 + x4 , 2! 3! 1 1 1 1 u3 (x) = 5 + x + x2 + x3 + x4 − x5 + · · · , 2! 3! 12 30 (5.219) 1 2 1 3 1 4 1 6 u4 (x) = 5 + x + x + x + x + x + · · · , 2! 3! 4! 90 1 2 1 3 1 4 1 u5 (x) = 5 + x + x + x + x + x5 + · · · , 2! 3! 4! 5! .. . . This gives 1 2 1 3 1 4 1 5 (5.220) un (x) = 4 + 1 + x + x + x + x + x + · · · . 2! 3! 4! 5!

5.3 Volterra Integro-Diﬀerential Equations of the First Kind

209

The exact solution is therefore given by u(x) = 4 + ex .

(5.221)

Example 5.31 Use the variational iteration method to solve the Volterra integro-diﬀerential equation of the ﬁrst kind x x 1 (x − t)u(t)dt + (x − t + 1)u (t)dt = sin x + x3 , u(0) = −1, u (0) = 1. 3! 0 0 (5.222) Diﬀerentiating both sides of this equation once with respect to x gives the Volterra integro-diﬀerential equation of the second kind x x 1 u (x) = cos x + x2 − u(t)dt − u (t)dt. (5.223) 2 0 0 The correction functional for equation this equation is given by x un+1 (x) = un (x) + ((t − x)Γ(t)) dt. (5.224) 0

where

t 1 (un (r) + un (r)) dr . un (t) − cos t − t2 + 2 0 The zeroth approximation u0 (x) can be selected by u0 (x) = −1 + x the initial conditions. This gives the successive approximations u0 (x) = −1 + x, 1 u1 (x) = x + x3 − cos x, 3! 1 1 u2 (x) = −1 + x + x2 − x4 + · · · , 2! 12 1 1 u3 (x) = −1 + x + x2 − x4 + · · · , 2! 4! 1 1 u4 (x) = −1 + x + x2 − x4 + · · · , 2! 4! 1 2 1 1 1 u5 (x) = −1 + x + x − x4 + x6 − x8 + · · · , 2! 4! 6! 8! .. . . This gives 1 2 1 4 1 6 1 8 un (x) = x − 1 − x + x − x + x + · · · . 2! 4! 6! 8! The exact solution is therefore given by Γ(t) =

u(x) = x − cos x.

(5.225) by using

(5.226)

(5.227)

(5.228)

210

5 Volterra Integro-Diﬀerential Equations

Example 5.32 Use the variational iteration method to solve the Volterra integro-diﬀerential equation of the ﬁrst kind x x sinh(x − t)u(t)dt + cosh(x − t)u (t)dt = xex , (5.229) 0 0 u(0) = u (0) = u (0) = 1. Diﬀerentiating both sides gives the Volterra integro-diﬀerential equation of the second kind u (x) = xex + ex −

x

0

x

cosh(x − t)u(t)dt −

0

sinh(x − t)u (t)dt. (5.230)

The correction functional for equation this equation is given by

1 x (t − x)2 Γ1 (t) dt, un+1 (x) = un (x) − (5.231) 2 0 where t t t t (t) − te − e + cosh(t − r)u (r)dr + u (r) dr . Γ1 (t) = u n n n 0

0

(5.232) The zeroth approximation can be selected by u0 (x) = 1 + x + 12 x2 by using the initial conditions. This gives the successive approximations 1 u0 (x) = 1 + x + x2 , 2! 1 1 1 1 u1 (x) = 1 + x + x2 + x3 + x4 + x5 + · · · , 2! 3! 4! 60 (5.233) 1 2 1 3 1 4 1 5 1 6 u2 (x) = 1 + x + x + x + x + x + x + · · · , 2! 3! 4! 5! 6! .. . . This gives 1 1 1 1 1 (5.234) un (x) = 1 + x + x2 + x3 + x4 + x5 + x6 + · · · . 2! 3! 4! 5! 6! The exact solution is therefore given by u(x) = ex .

(5.235)

Exercises 5.3.2 Solve the following Volterra integro-diﬀerential equations of the ﬁrst kind by using the variational iteration method: x 3 1 x 3 1. (x − t)u(t)dt − (x − t + 1)u (t)dt = + x − ex , u(0) = 0 2 2 2 0 0 x x (x − t)u(t)dt + (x − t + 1)u (t)dt = 1 + x − cos x, u(0) = 0 2. 0

0

References x 3. (x − t + 1)u (t)dt = 1 + sin x − cos x, u(0) = 0

0 x

4.

0 x

5. 0 x

6.

0 x

7. 0

8.

(x − t)u(t)dt +

x

0

(x − t + 1)u (t)dt =

211

1 2 1 3 x + x + sin x, u(0) = 1 2 6

(x − t + 1)u (t)dt = 2 sin x − x, u(0) = −1, u (0) = 1 (x − t)u(t)dt −

x

0

(x − t + 1)u (t)dt =

sinh(x − t)u(t)dt +

x

0

1 3 x − sinh x, u(0) = 1, u (0) = 1 6

cosh(x − t)u (t)dt =

1 x 1 xe + sinh x, 2 2

u(0) = 1, u (0) = −1 x x (x − t)u(t)dt + (x − t + 2)u (t)dt = 2x, u(0) = 0, u (0) = 1

0 x

9. 0

0

(x − t − 1)u(t)dt +

x

0

(x − t + 1)u (t)dt = 2x − 4 sin x,

u(0) = u (0) = 1, u (0) = −1 x x 10. (x − t + 1)u(t)dt − (x − t − 1)u (t)dt = 2x − 4 sin x, 0

0

u(0) = −1, u (0) = u (0) = 1 x x 11. sinh(x − t)u(t)dt − cosh(x − t)u (t)dt = −x + sinh x, 0

0

u(0) = u (0) = u (0) = 1 x 1 1 x 1 12. (x − t)u(t)dt − (x − t + 1)u (t)dt = − x + x3 , 2 2 6 0 0 u(0) = u (0) = 1, u (0) = 2

References 1. L.G. Chambers, Integral Equations, A Short Course, International Textbook Company, London, (1976). 2. R.F. Churchhouse, Handbook of Applicable Mathematics, Wiley, New York, (1981). 3. B.L. Moiseiwitsch, Integral Equations, Longman, London and New York, (1977). 4. V. Volterra, Theory of Functionals of Integral and Integro-Diﬀerential Equations, Dover, New York, (1959). 5. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, San Diego, (1986). 6. G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994). 7. G. Adomian and R. Rach, Noise terms in decomposition series solution, Comput. Math. Appl., 24 (1992) 61–64. 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 9. A.M.Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997).

212

5 Volterra Integro-Diﬀerential Equations

10. A.M. Wazwaz, The variational iteration method; a reliable tool for solving linear and nonlinear wave equations, Comput. Math. Appl., 54 (2007) 926–932. 11. P. Linz, A simple approximation method for solving Volterra integro-diﬀerential equations of the ﬁrst kind, J. Inst. Math. Appl., 14 (1974) 211–215. 12. P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, (1985).

Chapter 6

Fredholm Integro-Diﬀerential Equations

6.1 Introduction In Chapter 2, the conversion of boundary value problems to Fredholm integral equations was presented. However, the research work in this ﬁeld resulted in a new speciﬁc topic, where both diﬀerential and integral operators appeared together in the same equation. This new type of equations, with constant limits of integration, was termed as Fredholm integro-diﬀerential equations, given in the form b u(n) (x) = f (x) + K(x, t)u(t)dt, u(k) (0) = bk , 0 k n − 1, (6.1) a n

where u(n) (x) = ddxnu . Because the resulted equation in (6.1) combines the diﬀerential operator and the integral operator, then it is necessary to deﬁne initial conditions u(0), u (0), . . . , u(n−1) (0) for the determination of the particular solution u(x) of equation (6.1). Any Fredholm integro-diﬀerential equation is characterized by the existence of one or more of the derivatives u (x), u (x),. . . outside the integral sign. The Fredholm integro-diﬀerential equations of the second kind appear in a variety of scientiﬁc applications such as the theory of signal processing and neural networks [1–3]. A variety of numerical and analytical methods are used in the literature to solve the Fredholm integro-diﬀerential equations. In this chapter, these equations will be handled by using the traditional methods, namely, the direct computation method and the Taylor series method. Moreover, these equations will be handled by the newly developed methods, namely the variational iteration method, and the Adomian decomposition method.

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

214

6 Fredholm Integro-Diﬀerential Equations

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind In this chapter, we will focus our study on the equations that involve separable kernels where the kernel K(x, t) can be expressed as a ﬁnite sum of the form n gk (x) hk (t). (6.2) K(x, t) = k=1

Without loss of generality, we will make our analysis on a one term kernel K(x, t) of the form K(x, t) = g(x) h(t). (6.3) Other cases can be examined in a like manner. The non-separable kernel can be reduced to separable kernel by using the Taylor expansion for the kernel involved. However, the separable kernels only will be presented in this text. The methods that will be used were presented before in details, but here we will outline the main steps of each method to illustrate its use.

6.2.1 The Direct Computation Method The direct computation method has been introduced in Chapter 4. The standard form of the Fredholm integro-diﬀerential equation is given by b (n) K(x, t)u(t)dt, u(k) (0) = bk , 0 k n − 1 (6.4) u (x) = f (x) + a

where u(n) (x) indicates the nth derivative of u(x) with respect to x and bk are the initial conditions. Substituting (6.3) into (6.4) gives b (n) h(t)u(t)dt, u(k) (0) = bk , 0 k n − 1. (6.5) u (x) = f (x) + g(x) a

We can easily observe that the deﬁnite integral in the integro-diﬀerential equation (6.5) involves an integrand that depends completely on the variable t. This means that the deﬁnite integral at the right side of (6.5) is equivalent to a constant α. In other words, we set b α= h(t)u(t)dt. (6.6) a

Consequently, Equation. (6.5) becomes (6.7) u(n) (x) = f (x) + α g(x). Integrating both sides of (6.7) n times from 0 to x, and using the prescribed initial conditions, we can ﬁnd an expression for u(x) that involves the constant α in addition to the variable x. This means we can write u(x) = v(x; α).

(6.8)

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

215

Substituting (6.8) into the right side of (6.6), evaluating the integral, and solving the resulting equation, we determine a numerical value for the constant α. This leads to the exact solution u(x) obtained upon substituting the resulting value of α into (6.8). It is important to recall that this method leads always to the exact solution and not to series components. The method was used before in Chapter 4 for handling Fredholm integral equations. The method will be illustrated by studying the following examples. Example 6.1 Solve the following Fredholm integro-diﬀerential equation 1 u (x) = 3 + 6x + x tu(t)dt, u(0) = 0.

(6.9)

0

This equation may be written as u (x) = 3 + 6x + αx, obtained by setting

u(0) = 0,

(6.10)

1

α=

tu(t)dt.

(6.11)

0

Integrating both sides of (6.10) from 0 to x, and by using the given initial condition we obtain 1 (6.12) u(x) = 3x + 3x2 + αx2 . 2 Substituting (6.12) into (6.11) and evaluating the integral yield 1 7 1 α= (6.13) tu(t)dt = + α, 4 8 0 hence we ﬁnd α = 2. (6.14) The exact solution is given by u(x) = 3x + 4x2 .

(6.15)

Example 6.2 Solve the following Fredholm integro-diﬀerential equation 1 u(t)dt, u(0) = u (0) = 1. u (x) = 1 − e + ex +

(6.16)

0

This equation may be written as u (x) = 1 − e + ex + α, u(0) = 1, obtained by setting 1 α= u(t)dt.

u (0) = 1,

(6.17) (6.18)

0

Integrating both sides of (6.17) twice from 0 to x, and by using the given initial conditions we obtain

216

6 Fredholm Integro-Diﬀerential Equations

u(x) − x − 1 =

1−e+α 2 x + ex − 1 − x, 2

(6.19)

or equivalently

1−e+α 2 (6.20) x + ex . 2 Substituting (6.20) into (6.18) and evaluating the integral we obtain 1 1−e+α u(t)dt = α= + e − 1, (6.21) 6 0 hence we ﬁnd α = e − 1. (6.22) u(x) =

The exact solution is given by u(x) = ex .

(6.23)

Example 6.3 Solve the following Fredholm integro-diﬀerential equation π u (x) = 2+sin x− (x−t)u(t)dt, u(0) = 1, u (0) = 0, u (0) = −1. (6.24) 0

This equation may be written as u (x) = 2 + β − αx + sin x, u(0) = 1, u (0) = 0, u (0) = −1, obtained by setting π π u(t)dt, β = tu(t)dt. α= 0

(6.25)

(6.26)

0

Integrating both sides of (6.25) three times from 0 to x, and by using the given initial conditions we obtain 1 α u(x) = (2 + β)x3 − x4 + cos x. (6.27) 3! 4! Substituting (6.27) into (6.26) and evaluating the integrals we ﬁnd α α 6 π5 π4 (2 + β) − π5 , β = (2 + β) − π − 2. (6.28) α= 4! 5! 30 144 Solving this system of equations gives α = 0, β = −2. (6.29) The exact solution is given by u(x) = cos x.

(6.30)

Example 6.4 Solve the following Fredholm integro-diﬀerential equation π2 (iv) (x − 2t)u(t)dt, u (x) = (2x − π) + sin x + cos x − 0

u(0) = u (0) = 1, u (0) = u (0) = −1. This equation may be written as

(6.31)

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

u(iv) (x) = (2 − α)x + (2β − π) + sin x + cos x, u(0) = u (0) = 1, u (0) = u (0) = −1, obtained by setting π2 π2 α= u(t)dt, β = tu(t)dt. 0

217

(6.32)

(6.33)

0

Integrating both sides of (6.32) four times from 0 to x, and by using the given initial conditions we obtain 1 π 1 β − α x5 + − x4 . u(x) = sin x + cos x + (6.34) 60 120 12 24 Substituting (6.34) into (6.33) and evaluating the integrals we ﬁnd π5 π6 (10 + α) + (2 + β), α=− 46080 1920 (6.35) π7 π π6 β=− (29 + 3α) + ( + β). 322560 2 4608 Solving this system of equations gives π α = 2, β = . (6.36) 2 The exact solution is therefore given by u(x) = sin x + cos x, (6.37) obtained upon substituting (6.36) into (6.34).

Exercises 6.2.1 Solve the following Fredholm integro-diﬀerential equations by using the direct computation method 1 1. u (x) = 12x + u(t)dt, u(0) = 0 0

2. u (x) = 36x2 +

1

0

u(t)dt, u(0) = 1

3. u (x) = sec2 x − ln 2 + 4. u (x) = −10x +

1 −1

π 3

0

(x − t)u(t)dt, u(0) = 1

5. u (x) = 2 sec2 x tan x − 1 + 6. u (x) = −1 + cos x +

π 2

0

7. u (x) = −1 − sin x + 8. u (x) = 2 − cos x +

u(t)dt, u(0) = 0

0 π

0

π 2

π/4

0

u(t)dt, u(0) = 1

tu(t)dt, u(0) = 0 tu(t)dt, u(0) = 0, u (0) = 1

tu(t)dt, u(0) = 1, u (0) = 0

218

6 Fredholm Integro-Diﬀerential Equations

9. u (x) = −2x − sin x + cos x +

10. u (x) = 1 − 3 ln 2 + 4 cosh x +

π

11. u (x) = 2x − x sin x + 2 cos x + 12. u (x) = 4x − sin x +

xu(t)dt, u(0) = −1, u (0) = 1

0

ln 2

0

π 2

−π

(t − x)u(t)dt, u(0) = 0, u (0) = 0

2

π 2

−π

(x − t)2 u(t)dt, u(0) = 0, u (0) = 1

2

13. u (x) = 5 ln 2 − 3 − x + 4 cosh x + u(0) = u (0) = 0, u (0) = 4 14. u (x) = e − 2 − x + ex (3 + x) +

tu(t)dt, u(0) = 4, u (0) = 0

ln 2

0

1

0

(x − t)u(t)dt,

(x − t)u(t)dt,

u(0) = 0, u (0) = 1, u (0) = 2 π 2 tu(t)dt, u(0) = u (0) = 0, u (0) = 1, u (0) = −1 15. u(iv) (x) = −1 + sin x + (iv)

16. u

(x) = 4x + sin x +

0

π 2

−π

(x − t)2 u(t)dt, u(0) = u (0) = 0,

2

u (0) = 1, u (0) = −1

6.2.2 The Variational Iteration Method The variational iteration method [4] was used in Chapters 3, 4 and 5. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists. The standard ith order Fredholm integro-diﬀerential equation is of the form b

u(i) (x) = f (x) +

K(x, t)u(t)dt,

(6.38)

a i

where u(i) (x) = ddxui , and u(0), u (0), . . . , u(i−1) (0) are the initial conditions. The correction functional for the integro-diﬀerential equation (6.38) is x b un+1 (x) = un (x) + λ(ξ) u(i) K(ξ, r)˜ un (r)dr dξ. (6.39) n (ξ) − f (ξ) − 0

a

As presented before, the variational iteration method is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ(ξ) that can be identiﬁed optimally via integration by parts and by using a restricted variation. Having λ(ξ) determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1(x), n 0 of the solution u(x). The zeroth approxima-

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

219

tion u0 can be any selective function. However, using the given initial values u(0), u (0), . . . are preferably used for the selective zeroth approximation u0 as will be seen later. Consequently, the solution is given by u(x) = lim un (x). (6.40) n→∞

The VIM will be illustrated by studying the following examples. Example 6.5 Use the variational iteration method to solve the Fredholm integro-diﬀerential equation π2 tu(t)dt, u(0) = 0. (6.41) u (x) = −1 + cos x + 0

The correction functional for this equation is given by x π2 run (r) dr dξ, un+1 (x) = un (x) − un (ξ) + 1 − cos ξ −

(6.42)

0

0

where we used λ = −1 for ﬁrst-order integro-diﬀerential equations. Notice that the correction functional is the so-called Volterra-Fredholm integral equation, because it involves both types of integral equations. We can use the initial condition to select u0 (x) = u(0) = 0. Using this selection into the correction functional gives the following successive approximations u0 (x) = 0, x π2 u1 (x) = u0 (x) − ru0 (r)dr dξ u0 (ξ) + 1 − cos ξ − 0

0

= sin x − x, x u1 (ξ) + 1 − cos ξ − u2 (x) = u1 (x) − 0

0

π3 x , = (sin x − x) + x − 24 x u2 (ξ) + 1 − cos ξ − u3 (x) = u2 (x) − 0

π 2

π 2

0

ru1 (r)dr

dξ

ru2 (r)dr

(6.43) dξ

3 π π6 π3 x + x− x , = (sin x − x) + x − 24 24 576 x π2 u4 (x) = u3 (x) − ru3 (r)dr dξ u3 (ξ) + 1 − cos ξ −

0

0

3 6 π π6 π π3 x + x− x + x + ··· , = (sin x − x) + x − 24 24 576 576 and so on. The VIM admits the use of u(x) = lim un (x). n→∞

(6.44)

220

6 Fredholm Integro-Diﬀerential Equations

It is obvious that noise terms appear in the successive approximations, where by canceling these noise terms, we obtain the exact solution u(x) = sin x. (6.45) Example 6.6 Use the variational iteration method to solve the Fredholm integro-diﬀerential equation π u (x) = 2 − cos x + tu(t)dt, u(0) = 1, u (0) = 0. (6.46) 0

The correction functional for this equation is given by x π un+1 (x) = un (x)+ (ξ−x) un (ξ) − 2 + cos ξ − run (r) dr dξ, (6.47) 0

0

where λ = ξ − x for second-order integro-diﬀerential equations. We can use the initial conditions u(0) = 1, u (0) = 0 to select the zeroth approximation by u0 (x) = u(0) + xu (0) = 1. (6.48) Using this selection into the correction functional gives the following successive approximations u0 (x) = 1, x π u1 (x) = u0 (x) + (ξ − x) u0 (ξ) − 2 + cos ξ − ru0 (r)dr dξ, 0

0

π2 2 , = cos x + x 1 + 4 x (ξ − x) u1 (ξ) − 2 + cos ξ − u2 (x) = u1 (x) + 0

0

0

0

π

ru1 (r)dr dξ,

4 (6.49) π2 π2 π π6 2 2 2 −x 1+ +x + , = cos x + x 1 + 4 4 8 32 x π u3 (x) = u2 (x) + (ξ − x) u2 (ξ) − 2 + cos ξ − ru2 (r)dr dξ, 4 π2 π2 π π6 2 2 2 −x 1+ +x + = cos x + x 1 + 4 4 8 32 4 6 π π + + ··· , −x2 8 32 and so on. The VIM admits the use of u(x) = lim un (x). n→∞

Canceling the noise terms gives the exact solution u(x) = cos x.

(6.50) (6.51)

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

221

Example 6.7 Use the variational iteration method to solve the second-order Fredholm integro-diﬀerential equation π2 u (x) = 2x − x sin x + 2 cos x − (x − t)u(t)dt, u(0) = u (0) = 0. (6.52) −π 2

The correction functional for this equation is given by x un+1 (x) = un (x) + (ξ − x)Γn (ξ)dξ, n 0,

(6.53)

0

where Γn (ξ) = un (ξ) − 2ξ + ξ sin ξ − 2 cos ξ +

π 2

−π 2

(ξ − r)un (r) dr.

(6.54)

We can use the initial conditions u(0) = 0, u (0) = 0 to select the zeroth approximation by (6.55) u0 (x) = u(0) + xu (0) = 0. Using this selection into the correction functional gives the following successive approximations u0 (x) = 0, x 1 (ξ − x)Γ0 (ξ)dξ = x sin x + x3 , u1 (x) = u0 (x) + 3 0 x u2 (x) = u1 (x) + (ξ − x)Γ1 (ξ)dξ 0

π5 2 1 3 1 3 x , = x sin x + x + − x + 3 3 480 x u3 (x) = u2 (x) + (ξ − x)Γ2 (ξ)dξ

(6.56)

0

π5 2 1 3 π5 2 1 3 x + − x + ··· , = x sin x + x + − x + 3 3 480 480 and so on,where π2 Γi (ξ) = ui (ξ) − 2ξ + ξ sin ξ − 2 cos ξ + (ξ − r)ui (r) dr, i 0. (6.57) −π 2

Canceling the noise terms gives the exact solution u(x) = x sin x.

(6.58)

Example 6.8 Use the variational iteration method to solve the third-order Fredholm integro-diﬀerential equation 1 tu(t)dt, u(0) = 1, u (0) = 1, u (0) = 1. (6.59) u (x) = ex − 1 + 0

222

6 Fredholm Integro-Diﬀerential Equations

The correction functional for this equation is given by 1 1 x ξ un+1 (x) = un (x) − (ξ − x)2 u (ξ) − e + 1 − ru (r) dr dξ, n n 2 0 0 (6.60) where λ = − 12 (ξ − x)2 for third-order integro-diﬀerential equations. The zeroth approximation u0 (x) can be selected by 1 1 u0 (x) = u(0) + xu (0) + x2 u (0) = 1 + x + x2 . (6.61) 2! 2! Using this selection into the correction functional gives the following successive approximations 1 u0 (x) = 1 + x + x2 , 2! 1 1 x 2 ξ (ξ − x) u0 (ξ) − e + 1 − ru0 (r)dr dξ u1 (x) = u0 (x) − 2 0 0 1 3 x , = ex − 144 1 1 x 2 ξ (ξ − x) u1 (ξ) − e + 1 − ru1 (r)dr dξ (6.62) u2 (x) = u1 (x) − 2 0 0 1 3 29 3 1 3 x + x − x , = ex − 144 144 4320 1 1 x 2 ξ (ξ − x) u2 (ξ) − e + 1 − ru2 (r)dr dξ u3 (x) = u2 (x) − 2 0 0 1 3 29 3 1 3 29 3 x x + x − x + x + ··· , = e − 144 144 4320 4320 and so on. Canceling the noise terms gives the exact solution u(x) = ex .

(6.63)

Exercises 6.2.2 Solve the following Fredholm integro-diﬀerential equations by using the variational iteration method 1 u(t)dt, u(0) = 0 1. u (x) = 12x + 0

2. u (x) = −4 + 6x +

1

0

3. u (x) = sec2 x − ln 2 + 4. u (x) = −8x +

1

−1

u(t)dt, u(0) = 2

π 3

0

u(t)dt, u(0) = 0

(x − t)u(t)dt, u(0) = 2

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

5. u (x) = 2 sec2 x tan x − 1 + 6. u (x) = −1 + cos x +

7. u (x) = −1 − sin x + 8. u (x) = −2 − cos x +

π 2

π 2

u(t)dt, u(0) = 1

0

tu(t)dt, u(0) = 0, u (0) = 1

0

π/4

(2 − t)u(t)dt, u(0) = 0

0

π

(1 − t)u(t)dt, u(0) = 1, u (0) = 0

0

9. u (x) = −2x − sin x + cos x +

π

0

10. u (x) = −4 + 3 ln 2 + 4 cosh x + 11. u (x) = 2x − x sin x + 2 cos x + 12. u (x) = 4π − cos x +

223

xu(t)dt, u(0) = −1, u (0) = 1

ln 2 0 π 2

−π

(1 − t)u(t)dt, u(0) = 4, u (0) = 0

(t − x)u(t)dt, u(0) = 0, u (0) = 0

2

π

−π

(x − t)2 u(t)dt, u(0) = 1, u (0) = 0

13. u (x) = 5 ln 2 − 3 − x + 4 cosh x +

ln 2

0

(x − t)u(t)dt,

u(0) = u (0) = 0, u (0) = 4 1 tu(t)dt, u(0) = u (0) = u (0) = 1 14. u (x) = −1 + ex + 0

(iv)

15. u

(x) = −1 + sin x +

16. u(iv) (x) = 2 + cos x +

0 π

0

π 2

tu(t)dt, u(0) = u (0) = 0, u (0) = 1, u (0) = −1

tu(t)dt, u(0) = 1, u (0) = u (0) = 0, u (0) = −1

6.2.3 The Adomian Decomposition Method The Adomian decomposition method was presented before and used thoroughly in previous chapters. This method was used in its standard form to obtain the solution in an inﬁnite series of components that can be recurrently determined. In addition, it was used jointly with the noise terms phenomenon. Moreover, a modiﬁed form was developed to facilitate the computational work. The obtained series may give the exact solution if such a solution exists. Otherwise, the series gives an approximation for the solution that gives high accuracy level. The main focus in this section will be directed to converting a Fredholm integro-diﬀerential equation to an equivalent Fredholm integral equation. The converted Fredholm integral equation can then be solved by any of the methods presented before in Chapter 4, such as the direct computation method, successive approximations method, and others. However, in this section we

224

6 Fredholm Integro-Diﬀerential Equations

will use the Adomian decomposition method for solving the Fredholm integrodiﬀerential equation by converting it ﬁrst to an integral equation. The Adomian decomposition method was presented in details in previous chapters, only a summary will be outlined here. Without loss of generality, we may assume a second order Fredholm integro-diﬀerential equation given by b u (x) = f (x) + K(x, t)u(t)dt, u(0) = a0 , u (0) = a1 . (6.64) a

Integrating both sides of (6.64) from 0 to x twice gives a −1 −1 K(x, t) u(t)dt , u(x) = a0 + a1 x + L (f (x)) + L

(6.65)

a

where the initial conditions u(0) = a0 and u (0) = a1 are used, and L−1 is a two-fold integral operator. The Adomian decomposition method admits the use of the decomposition series ∞ u(x) = un (x), (6.66) n=0

into both sides of (6.65) to obtain ∞ un (x) = a0 + a1 x + L−1 (f (x)) + L−1 n=0

x

K(x, t) 0

∞

un (t)dt , (6.67)

n=0

or equivalently u0 (x) + u1 (x) + u2 (x) + u3 (x) + · · · = a0 + a1 x + L−1 (f (x)) b b −1 −1 K(x, t)u0 (t)dt + L K(x, t)u1 (t)dt +L a

+L−1

a

b

K(x, t)u2 (t)dt

(6.68)

+ ··· .

a

Consequently, to determine the components u0 (x), u1 (x), u2 (x), u3 (x), . . . of the solution u(x), we set the recurrence relation u0 (x) = a0 + a1 x + L−1 (f (x)) , b (6.69) uk+1 (x) = L−1 K(x, t)uk (t)dt , k 0, a

where the zeroth component u0 (x) is deﬁned by all terms not included inside the integral sign of (6.68). Having determined the components ui (x), i 0, the solution u(x) of (6.64) is then obtained in a series form. Using (6.66), the obtained series converges to the exact solution if such a solution exists. We point out that some modiﬁed forms of the Adomian method was developed in the literature. The most commonly used form is that one developed in [5–6] and was employed in Chapter 3. In what follows, we summarize the main steps for this modiﬁed form.

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

225

The Modiﬁed Decomposition Method The Adomian decomposition method provides the solutions in an inﬁnite series of components. The method substitutes the decomposition series of u(x), given by ∞ un (x), (6.70) u(x) = n=0

into both sides of the Fredholm integral equation b u(x) = f (x) + λ K(x, t)u(t)dt.

(6.71)

a

The standard Adomian decomposition method introduces the recurrence relation u0 (x) = f (x), b (6.72) K(x, t)uk (t)dt, k 0. uk+1 (x) = λ a

In view of (6.72), the components un (x), n 0 are readily obtained. The modiﬁed decomposition method presents a slight variation to the recurrence relation (6.72) to determine the components of u(x) in an easier and faster manner. For many cases, the function f (x) can be set as the sum of two partial functions, namely f1 (x) and f2 (x). In other words, we can set (6.73) f (x) = f1 (x) + f2 (x). In view of (6.73), we introduce a qualitative change in the formation of the recurrence relation (6.72). The modiﬁed decomposition method identiﬁes the zeroth component u0 (x) by one part of f (x), namely f1 (x) or f2 (x). The other part of f (x) can be added to the component u1 (x) that exists in the standard recurrence relation. The modiﬁed decomposition method admits the use of the modiﬁed recurrence relation u0 (x) = f1 (x), b K(x, t)u0 (t)dt, u1 (x) = f2 (x) + λ (6.74) a b uk+1 (x) = λ K(x, t)uk (t)dt, k 1. a

It is obvious that the diﬀerence between the standard recurrence relation (6.72) and the modiﬁed recurrence relation (6.74) rests only in the formation of the ﬁrst two components u0 (x) and u1 (x) only. The other components uj , j 2 remain the same in the two recurrence relations. It is interesting to recall that the noise terms phenomenon can be used here. This phenomenon was employed before, but it is useful to summarize the main steps of this phenomenon.

226

6 Fredholm Integro-Diﬀerential Equations

The Noise Terms Phenomenon It was shown before that if the noise terms appear between components of u(x), then the exact solution can be obtained only by considering only the ﬁrst two components u0 (x) and u1 (x). The noise terms are deﬁned as the identical terms, with opposite signs, that may appear between the components of the solution u(x). The conclusion made by [5–8] suggests that if we observe the appearance of identical terms in both components with opposite signs, then by canceling these terms, the remaining non-canceled terms of u0 may in some cases provide the exact solution, that should be justiﬁed through substitution. It was formally proved that other terms in other components will vanish in the limit if the noise terms occurred in u0 (x) and u1 (x). However, if the exact solution was not attainable by using this phenomenon, then we should continue determining other components of u(x) in a standard way. The Adomian decomposition method, the noise terms phenomenon, and the modiﬁed decomposition method for solving Fredholm integro-diﬀerential equations will be illustrated by studying the following examples. The selected equations are of orders 1, 2, 3, and 4. Other equations of higher orders can be treated in a like manner. Example 6.9 Use the Adomian decomposition method to solve the Fredholm integrodiﬀerential equation 1 2 u (x) = 36x + u(t)dt, u(0) = 1. (6.75) 0

Recall that the integral at the right side is equivalent to a constant because it depends only on the variable t with constant limits of integration for t. Integrating both sides of Eq. (6.75) from 0 to x gives 1 u(t)dt , (6.76) u(x) − u(0) = 12x3 + x 0

which gives upon using the initial condition 3 u(x) = 1 + 12x + x

1

u(t)dt .

(6.77)

0

Substituting the series assumption u(x) =

∞

un (x),

(6.78)

n=0

into both sides of (6.77) gives ∞ 3 un (x) = 1 + 12x + x n=0

∞ 1 0 n=0

un(t)dt .

(6.79)

The components uj (x), j 0 of u(x) can be determined by using the recurrence relation

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

u0 (x) = 1 + 12x3 ,

uk+1 (x) = x

1 0

uk (t)dt,

This in turn gives

3

u0 (x) = 1 + 12x , u2 (x) = x

u1 (x) = x

1 0

u1 (t)dt = 2x,

u3 (x) = x

k 0.

227

(6.80)

1

u0 (t)dt = 4x,

0 1 0

u2 (t)dt = x,

.. . The solution in a series form is given by 1 1 3 u(x) = 1 + 12x + 4x 1 + + + · · · , 2 4 which gives the exact solution u(x) = 1 + 8x + 12x3 , obtained upon evaluating the sum of the inﬁnite geometric series.

(6.81)

(6.82)

(6.83)

Example 6.10 Use the Adomian decomposition method to solve the Fredholm integrodiﬀerential equation π u (x) = 2 − cos x + tu(t)dt, u(0) = 1, u (0) = 0. (6.84) 0

Integrating both sides of Eq. (6.84) twice from 0 to x gives π 1 2 2 tu(t)dt , u(x) − u(0) − xu (0) = −1 + cos x + x + x 2 0 which gives upon using the initial conditions π 1 2 2 u(x) = cos x + x + x tu(t)dt . 2 0 Substituting the series assumption ∞ un (x), u(x) =

(6.85)

(6.86)

(6.87)

n=0

into both sides of (6.86) gives ∞ ∞ π 1 un (x) = cos x + x2 + x2 t un (t)dt . 2 0 n=0 n=0 Proceeding as before, we set the recurrence relation π 1 u0 (x) = cos x + x2 , uk+1 (x) = x2 tuk (t)dt, k 0. 2 0 This in turn gives 1 2 π π4 2 2 x . u0 (t)dt = −x2 + u0 (x) = cos x + x , u1 (x) = x 2 8 0

(6.88)

(6.89)

(6.90)

228

6 Fredholm Integro-Diﬀerential Equations

The noise terms ±x2 appear between u0 (x) and u1 (x). Canceling the noise term from u0 (x) gives the exact solution u(x) = cos x, (6.91) that justiﬁes the integro-diﬀerential equation. Example 6.11 Use the Adomian decomposition method to solve the Fredholm integrodiﬀerential equation 1 u (x) = ex − x + xtu(t)dt, u(0) = u (0) = u (0) = 1. (6.92) 0

Integrating both sides of Eq. (6.92) three times from 0 to x, and using the initial conditions we obtain 1 1 3 1 3 x u(x) = e − x + x tu(t)dt . (6.93) 3! 3! 0 Proceeding as before, we set the recurrence relation 1 1 1 tuk (t)dt, k 0. (6.94) u0 (x) = ex − x3 , uk+1 (x) = x3 3! 3! 0 This in turn gives 1 u0 (x) = ex − x3 , 3! 1 1 29 3 u1 (x) = x3 tu0 (t)dt = x , 3! 180 0 1 1 29 3 (6.95) tu1 (t)dt = u2 (x) = x3 x , 3! 5400 0 1 1 29 u3 (x) = x3 x3 , tu2 (t)dt = 3! 162000 0 .. . The solution in a series form is given by 1 29 3 1 1 u(x) = ex − x3 + x 1+ + + ··· . 3! 180 30 900 The inﬁnite geometric series has a1 = 1, r = geometric series is given by 30 1 . S= 1 = 29 1 − 30 Using this result gives the exact solution u(x) = ex

1 3.

(6.96)

The sum of the inﬁnite (6.97)

(6.98)

Example 6.12 Use the Adomian decomposition method to solve the Fredholm integrodiﬀerential equation

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

u(iv) (x) = 1 − x + sin x +

u(0) = u (0) = 0,

0

π 2

(x − t)u(t)dt,

229

(6.99)

u (0) = −u (0) = 1.

Integrating both sides of Eq. (6.99) four times from 0 to x and using the initial conditions we ﬁnd π π 1 4 1 4 2 1 3 2 1 3 u(t)dt − x tu(t)dt. (6.100) u(x) = sin x + x − x + x 3! 4! 4! 3! 0 0 Substituting the series assumption and proceeding as before we obtain the recurrence relation 1 1 u0 (x) = sin x + x3 − x4 , 3! 4! π π2 (6.101) 1 4 2 1 uk+1 = x uk (t)dt − x3 tuk (t)dt, k 0. 4! 3! 0 0 This in turn gives 1 1 1 1 u0 (x) = sin x + x3 − x4 , u1 (x) = − x3 + x4 + · · · , 3! 4! 3! 4! (6.102) .. . 1 3 1 4 x , − 4! x 3!

Canceling the noise terms

from u0 (x) gives the exact solution

u(x) = sin x.

(6.103)

Exercises 6.2.3 Solve the following Fredholm integro-diﬀerential equations by using the Adomian decomposition method π 3 u(t)dt, u(0) = 0 1. u (x) = sec2 x − ln 2 + 2. u (x) = −1 + 24x + 3. u (x) = 6 + 17x +

0

1 0

u(t)dt, u(0) = 0

1 0

xu(t)dt, u(0) = 0

√ 4. u (x) = − 3 + 2 sec2 x tan x + π 5. u (x) = − x + sin 2x + 2 6. u (x) = 1 − x + cos x + 7. u (x) = −

π 2

−π

π 3

0

u(t)dt, u(0) = 1

u(t)dt, u(0) = 0

2

π 2

0

π − 2 cos 2x + 4

(x − t)u(t)dt, u(0) = 0

0

π 2

u(t)dt, u(0) = 1, u (0) = 0

230

6 Fredholm Integro-Diﬀerential Equations

8. u (x) = 2x − cos x +

π

xtu(t)dt, u(0) = 1, u (0) = 0

0

9. u (x) = −2x − sin x + cos x + 10. u (x) = −2 − cos x +

π 0

π

0

xu(t)dt, u(0) = −1, u (0) = 1

(x − t)u(t)dt, u(0) = 1, u (0) = 0

11. u (x) = 3 ln 2 − 1 − 3x + 4 cosh x +

ln 2 0

(x − t)u(t)dt, u(0) = 4, u (0) = 0

1 1 1 + (x − t)u(t)dt, u(0) = 0, u (0) = 1 + (1 − 2 ln 2)x − 4 (1 + x)2 0 1 13. u (x) = 2e−1 − e−x + tu(t)dt, u(0) = u (0) = 1, u (0) = −1 12. u (x) =

−1

14. u (x) = e − 2 − x + (3 + x)ex + 15. u(iv) (x) = −2x + sin x + cos x +

1

0 π 2

−π

(x − t)u(t)dt, u(0) = u (0) = 1, u (0) = 2 xtu(t)dt,

2

u(0) = u (0) = 1, u (0) = u (0) = −1 16. u(iv) (x) =

6 1 + (1 − 2 ln 2)x − + 4 (1 + x)4

1 0

(x − t)u(t)dt,

u(0) = 0, u (0) = 1, u (0) = −1, u (0) = 2

6.2.4 The Series Solution Method The series solution method was used before, where the generic form of Taylor series for an analytic solution u(x) at x = 0 ∞ a n xn (6.104) u(x) = n=0

is used. Following the discussion presented before in Chapters 3 and 4, the series solution method will be used for solving Fredholm integro-diﬀerential equations. We will assume that the solution u(x) of the Fredholm integrodiﬀerential equation b u(k) (x) = f (x) + λ K(x, t)u(t)dt, u(j) (0) = aj , 0 j (k − 1), (6.105) a

is analytic, and therefore possesses a Taylor series of the form given in (6.104), where the coeﬃcients an will be determined recurrently. Substituting (6.104) into both sides of (6.105) gives

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

∞

(k) n

an x

b

= T (f (x)) + λ

K(x, t) a

n=0

∞

231

an t

n

dt,

(6.106)

n=0

or for simplicity we use

(k) 2 a0 + a1 x + a2 x + · · · = T (f (x))+λ

b

K(x, t) a0 + a1 t + a2 t2 + · · · dt,

a

(6.107) where T (f (x)) is the Taylor series for f (x). The integral equation (6.105) will be converted to a traditional integral in (6.106) or (6.107) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. Moreover, the given initial equations should be used in the series assumption (6.104). We ﬁrst integrate the right side of the integral in (6.106) or (6.107), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x in both sides of the resulting equation to obtain a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (6.104). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. It is worth noting that using the series solution method for solving Fredholm integro-diﬀerential equations gives exact solutions if the solution u(x) is a polynomial. However, if the solution is any other elementary function such as sin x, ex , etc, the series method gives the exact solution after rounding few of the coeﬃcients aj , j 0. This will be illustrated by studying the following examples. Example 6.13 Solve the Fredholm integro-diﬀerential equation by using the series solution method 1 (1 − xt)u(t)dt, u(0) = 1. (6.108) u (x) = 4 + 4x + −1

Substituting u(x) by the series

u (x) =

∞

an x

n

,

(6.109)

n=0

into both sides of Eq. (6.108) leads to 1 ∞ ∞ n−1 n (1 − xt) dt. nan x = 4 + 4x + an t n=1

−1

n=0

(6.110)

232

6 Fredholm Integro-Diﬀerential Equations

Notice that a0 = 1 from the given initial conditions. Evaluating the integral at the right side gives a1 + 2a2 x + 3a3 x2 + · · · (6.111) 2 2 2 2 2 2 2 2 = 6 + a2 + a4 + a6 + a8 + (4 − a1 − a3 − a5 − a7 )x. 3 5 7 9 3 5 7 9 Equating the coeﬃcients of like powers of x in both sides of (6.111) gives (6.112) a1 = 6, an = 0, n 2. The exact solution is given by u(x) = 1 + 6x,

(6.113)

where we used a0 = 1 from the initial condition. Example 6.14 Solve the Fredholm integro-diﬀerential equation by using the series solution method 1 2 (x − t)u(t)dt, u(0) = 1. (6.114) u (x) = 7 − 2x + 15x + −1

Substituting the series

u (x) =

∞

an x

n

,

(6.115)

n=0

into both sides of Eq. (6.114) leads to ∞ nan xn−1 = 7 − 2x + 15x2 + n=1

1

−1

(x − t)

∞

an t

n

dt.

(6.116)

n=0

Notice that a0 = 1 from the given initial conditions. Evaluating the integral at the right side gives 2 2 2 2 a1 + 2a2 x + 3a3 x2 + · · · = 7 − a1 − a3 − a5 − a7 3 5 7 9 (6.117) 2 2 2 2 + a2 + a4 + a6 + a8 x + 15x2 . 3 5 7 9 Equating the coeﬃcients of like powers of x in both sides of (6.117) gives a1 = 3, a2 = 0, The exact solution is given by

a3 = 5,

an = 0,

u(x) = 1 + 3x + 5x3 , where we used a0 = 1 from the initial condition.

n 4.

(6.118) (6.119)

Example 6.15 Solve the Fredholm integro-diﬀerential equation by using the series solution method 1 5 (xt2 − x2 t)u(t)dt, u(0) = 1, u (0) = 1. (6.120) u (x) = − 11x + 3 −1

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

Substituting the series

u (x) =

∞

233

an x

n

,

(6.121)

n=0

into both sides of Eq. (6.120) leads to 1 ∞ ∞ 5 n−2 2 2 n (xt − x t) n(n − 1)an x = − 11x + an t dt. 3 −1 n=2 n=0

(6.122)

Notice that a0 = a1 = 1 from the given initial conditions. Evaluating the integral at the right side gives 31 2 5 2 2 2 2 2a2 + 6a3 x + 12a4 x + · · · = + − + a2 + a4 + a6 + a8 x 3 7 5 7 9 11 2 2 2 2 − + a3 + a5 + a7 x2 . 3 5 7 9 (6.123) Equating the coeﬃcients of like powers of x in both sides of (6.123) gives 5 5 (6.124) a2 = , a3 = − , an = 0, n 4. 6 3 The exact solution is given by 5 5 u(x) = 1 + x + x2 − x3 , (6.125) 6 3 where we used a0 = a1 = 1 from the initial condition. Example 6.16 Solve the Fredholm integro-diﬀerential equation by using the series solution method π u (x) = 1 − x − cos x + xtu(t)dt, u(0) = 0, u (0) = 1, u (0) = 0. (6.126) 0

Substituting the series

u (x) =

∞

an x

n

,

(6.127)

n=0

into both sides of Eq. (6.126) leads to ∞ n−3 n(n − 1)(n − 2)an x = 1 − x − cos x +

0

n=3

π

xt

∞

an t

n

dt. (6.128)

n=0

Evaluating the integral at the right side and proceeding as before we obtain (−1)n (6.129) a2n+1 = , a2n+2 = 0, n 1. (2n + 1)! Recall that the initial conditions give a0 = 0, a1 = 1, a2 = 0. Combining the initial conditions with the last results for the coeﬃcients we ﬁnd u(x) = sin x.

(6.130)

234

6 Fredholm Integro-Diﬀerential Equations

It is to be noted that we used terms up to O(x12 ) in the series to obtain this result.

Exercises 6.2.4 Use the series solution method to solve the following Fredholm integro-diﬀerential equations: 1 1. u (x) = 4x + (x − t)u(t)dt, u(0) = 2 2. u (x) = 4 +

−1

1 −1

(1 − x2 t2 )u(t)dt, u(0) = −2

3. u (x) = 5 + 4x +

1

−1

(x − t)u(t)dt, u(0) = 0

1 10 2 (x2 − t2 )u(t)dt, u(0) = 0 x + 3 −1 1 92 5. u (x) = (x − t)u(t)dt, u(0) = u (0) = 0 + 5 −1 1 6. u (x) = −18 + (x − t)u(t)dt, u(0) = 3, u (0) = 0

4. u (x) = 10x −

−1

7. u (x) = −18 + 8. u (x) = 32x +

1

−1 1

−1

(1 − xt)u(t)dt, u(0) = 3, u (0) = 0

(1 − xt)u(t)dt, u(0) = 1, u (0) = 0

9. u (x) = −2x + sin x − cos x + 10. u (x) = 2 + sin x +

π 0

11. u (x) = −1 + sin x + 12. u (x) = 1 − e + ex +

π 0

xu(t)dt, u(0) = u (0) = 1, u (0) = −1

tu(t)dt, u(0) = 1, u (0) = 0, u (0) = −1

π 2

0 1 0

u(t)dt, u(0) = 1, u (0) = 0, u (0) = −1 u(t)dt, u(0) = u (0) = u (0) = 1

References 1. H.T. Davis, Introduction to Nonlinear Diﬀerential and Integral Equations, Dover, New York, (1962). 2. R.P. Kanwal, Linear Integral Equations, Birkhauser, Boston, (1997). 3. G. Micula and P. Pavel, Diﬀerential and Integral Equations through Practical Problems and Exercises, Kluwer, Boston, (1992). 4. J.H. He, Variational iteration method for autonomous ordinary diﬀerential systems, Appl. Math. Comput., 114(2/3) (2000) 115–123.

References

235

5. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 6. A.M. Wazwaz, A reliable modiﬁcation of the Adomian decomposition method, Appl. Math. Comput., 102 (1999) 77–86. 7. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, San Diego, (1986). 8. G. Adomian, and R. Rach, Noise terms in decomposition series solution, Comput. Math. Appl., 24 (1992) 61–64.

Chapter 7

Abel’s Integral Equation and Singular Integral Equations

7.1 Introduction Abel’s integral equation occurs in many branches of scientiﬁc ﬁelds [1], such as microscopy, seismology, radio astronomy, electron emission, atomic scattering, radar ranging, plasma diagnostics, X-ray radiography, and optical ﬁber evaluation. Abel’s integral equation is the earliest example of an integral equation [2]. In Chapter 2, Abel’s integral equation was deﬁned as a singular integral equation. Volterra integral equations of the ﬁrst kind h(x) f (x) = λ K(x, t)u(t)dt, (7.1) g(x)

or of the second kind

h(x)

K(x, t)u(t)dt,

u(x) = f (x) + λ

(7.2)

g(x)

are called singular [3–4] if: 1. one of the limits of integration g(x), h(x) or both are inﬁnite, or 2. if the kernel K(x, t) becomes inﬁnite at one or more points at the range of integration. Examples of the ﬁrst type are given by Fourier transform and Laplace transform of the function u(x) ∞ e−ıλx u(x)dx, (7.3) F (λ) = −∞

L{u(x)}(s) =

∞

e−sx u(x)dx.

(7.4)

0

respectively. It is obvious that the range of integration is inﬁnite for the Fourier transform and Laplace transform respectively. Examples of the second type are the Abel’s integral equation, generalized Abel’s integral equation, and weakly singular integral equation are given by A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

238

7 Abel’s Integral Equation and Singular Integral Equations

f (x) =

0 x

f (x) = 0

1 u(t)dt, (x − t)

1 u(t)dt, 0 < α < 1, (x − t)α

and

x

u(x) = f (x) + 0

x

1 u(t)dt, 0 < α < 1, (x − t)α

(7.5) (7.6)

(7.7)

respectively. It is clear that the kernel in each equation becomes inﬁnite at the upper limit t = x. The last three singular integral equations are among the earliest integral equations established by the Norwegian mathematician Niles Abel in 1823. In this chapter we will focus our study on the second style of singular integral equations, namely the equations where the kernel K(x, t) becomes unbounded at one or more points of singularities in its domain of deﬁnition. The equations that will be investigated are Abel’s problem, generalized Abel integral equations and the weakly-singular second-kind Volterra type integral equations. In a manner parallel to the approach used in previous chapters, we will focus our study on the techniques that will guarantee the existence of a unique solution to any singular integral equation. We point out here that singular integral equations are in general very diﬃcult to handle.

7.2 Abel’s Integral Equation Abel in 1823 investigated the motion of a particle that slides down along a smooth unknown curve, in a vertical plane, under the inﬂuence of the gravity. The particle takes the time f (x) to move from the highest point of vertical height x to the lowest point 0 on the curve. The Abel’s problem is derived to ﬁnd the equation of that curve. Abel derived the equation of motion of the sliding particle along a smooth curve by the singular integral equation x 1 √ f (x) = u(t)dt, (7.8) x −t 0 where f (x) is a predetermined data function, and u(x) is the solution that will be determined. It is to be noted that Abel’s integral equation (7.8) is also called Volterra integral equation of the ﬁrst kind. Besides the kernel K(x, t) in Abel’s integral equation (7.8) is 1 , (7.9) K(x, t) = √ x−t where K(x, t) → ∞, as t → x. (7.10)

7.2 Abel’s Integral Equation

239

It is interesting to point out that although Abel’s integral equation is a Volterra integral equation of the ﬁrst kind, but two of the methods used before in Section 3.3, namely the series solution method and the conversion to a second kind Volterra equation, are not applicable here. The series solution cannot be used in this case especially if u(x) is not analytic. Moreover, converting Abel’s integral equation to a second kind Volterra equation is not obtainable because we cannot use Leibnitz rule due to the singularity behavior of the kernel in (7.8).

7.2.1 The Laplace Transform Method Although the Laplace transform method was presented before, but a brief summary will be helpful. In the convolution theorem for the Laplace transform method, it was stated that if the kernel K(x, t) of the integral equation x f (x) = K(x, t)u(t)dt, (7.11) 0

depends on the diﬀerence x−t, then it is called a diﬀerence kernel. The Abel’s integral equation can thus be expressed as x K(x − t)u(t)dt. (7.12) f (x) = 0

Consider two functions f1 (x) and f2 (x) that possess the conditions needed for the existence of Laplace transform for each. Let the Laplace transforms for the functions f1 (x) and f2 (x) be given by L{f1 (x)} = F1 (s), (7.13) L{f2 (x)} = F2 (s). The Laplace convolution product of these two functions is deﬁned by x f1 (x − t)f2 (t)dt, (7.14) (f1 ∗ f2 )(x) =

or (f2 ∗ f1 )(x) =

0

x 0

f2 (x − t)f1 (t)dt.

(7.15)

Recall that (f1 ∗ f2 )(x) = (f2 ∗ f1 )(x).

(7.16)

We can easily show that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by L{(f1 ∗ f2 )(x)} = F1 (s)F2 (s). (7.17) Based on this summary, we will examine Abel’s integral equation where the kernel is a diﬀerence kernel. Recall that we will apply the Laplace transform method and the inverse of the Laplace transform using Table 2 in section 1.5. Taking Laplace transforms of both sides of (7.8) leads to

240

7 Abel’s Integral Equation and Singular Integral Equations 1

L{f (x)} = L{u(x)}L{x− 2 }, or equivalently F (s) = U (s)

(7.18)

√ π Γ(1/2) = U (s) , 1/2 1/2 s s

(7.19)

that gives

s1/2 (7.20) U (s) = √ F (s), π √ where Γ is the gamma function, and Γ( 12 ) = π. In Appendix D, the deﬁnition of the gamma function and some of the relations related to it are given. The last equation (7.20) can be rewritten as 1 s √ (7.21) U (s) = ( πs− 2 F (s)), π which can be rewritten by s (7.22) L{u(x)} = L{y(x)}, π where x 1 (x − t)− 2 f (t)dt. (7.23) y(x) = 0

Using the fact

L{y (x)} = sL{y(x)} − y(0),

(7.24)

into (7.22) we obtain L{u(x)} =

1 L{y (x)}. π

(7.25)

Applying L−1 to both sides of (7.25) gives the formula x f (t) 1 d √ dt, (7.26) u(x) = π dx 0 x−t that will be used for the determination of the solution u(x). Notice that the formula (7.26) will be used for solving Abel’s integral equation, and it is not necessary to use Laplace transform method for each problem. Abel’s problem given by (7.8) can be solved directly by using the formula (7.26) where the unknown function u(x) has been replaced by the given function f (x). For f (x) = xn , n is a positive integer, Table 7.1 can be used for evaluating the integral in Abel’s problem. Table 7.1 For integrals of the form u(x) =

x 0

n

√t dt, n x−t

0. 1

u(x)

√ 2c x

4 3 x2 3

16 5 x2 15

32 7 x2 35

.. .

2n+1 Γ(n + 1)xn+ 2 1 · 3 · 5 · · · (2n + 1)

f (t)

c

t

t2

t3

.. .

tn

7.2 Abel’s Integral Equation

241 n

However, for f (x) = x 2 , n is an odd positive integer, Table 7.2 can be used for evaluating the integral in Abel’s problem. Table 7.2 For integrals of the form u(x) = u(x)

√ 2c x

1 πx 2

f (t)

c

t2

1

3 2 πx 8 3

t2

x 0

n

2 √t dt. x−t

5 πx3 16 5

t2

.. .

Γ( n+2 ) √ n+1 2 πx 2 Γ( n+3 ) 2

.. .

n

t2

More details can be found in Appendix B. Using formula (7.26) to determine the solution of Abel’s problem (7.8) will be illustrated by the following examples. Example 7.1 Solve the following Abel’s integral equation x √ 1 √ u(t)dt. 2π x = x−t 0 √ Substituting f (x) = 2π x in (7.26) gives √ x 1 d 2π t d √ u(x) = (πx) = π. dt = π dx 0 dx x−t

(7.27)

(7.28)

Example 7.2 Solve the following Abel’s integral equation x 1 4 32 √ x = u(t)dt. 3 x −t 0

(7.29)

3

Substituting f (x) = 43 x 2 in (7.26) gives x 4 3 t2 1 d 1 d 2 √3 u(x) = (x ) = x dt = π dx 0 2 dx x−t

(7.30)

Example 7.3 Solve the following Abel’s integral equation x 8 3 16 5 1 √ u(t)dt. x2 + x2 = 3 5 x−t 0 Using (7.26) gives x 8 3 16 5 2 2 1 d 3t + 5 t √ u(x) = dt, π dx 0 x−t 1 d (πx2 + πx3 ) = 2x + 3x2 . = π dx

(7.31)

(7.32)

242

7 Abel’s Integral Equation and Singular Integral Equations

Example 7.4 Find two-terms approximation of the solution of the following Abel’s integral equation x 1 √ sin x = u(t)dt, x ∈ [0, 1]. (7.33) x−t 0 Using (7.26) gives

x 1 3 t − 3! t sin t 1 d √ √ dt = dt, π dx x − t x − t 0 0 16 7 8 5 1 1√ 1 d 4 3 2 2 2 x− x = . x − x = π dx 3 105 π 2 15

u(x) =

1 d π dx

x

(7.34)

Exercises 7.2.1 In Exercises 1–12, solve the following Abel’s integral equations where x ∈ [0, 2] x x 1 1 π √ √ u(t)dt 2. x = u(t)dt 1. π = x−t 2 x−t 0 0 x x √ √ 8 3 1 1 √ √ u(t)dt 4. 2 x + x 2 = u(t)dt 3. 2 x = x − t 3 x −t 0 0 x x 3 π 1 1 √ √ u(t)dt 6. πx2 = u(t)dt 5. (x2 − x) = 2 x−t 8 x−t 0 0 x x 1 4 3 4 3 1 1 √ √ u(t)dt 8. πx + x 2 = u(t)dt 7. x 2 = 3 x−t 2 3 x−t 0 0 x x √ 1 1 1 √ √ u(t)dt 10. x3 = u(t)dt 9. 2 x − πx = 2 x − t x −t 0 0 x x 3 1 1 √ √ u(t)dt 12. 3x2 = u(t)dt 11. 4x 2 − x = x−t x−t 0 0 In Exercises 13–16, ﬁnd two-terms approximation for the solution of the following Abel’s integral equations x x 1 1 13. π sinh x = √ √ u(t)dt 14. π sinh x ln(1 + x) = u(t)dt x−t x−t 0 0 x x 1 1 √ √ u(t)dt 16. π sin x tan x = u(t)dt, x 15. π cosh x ln(1 + x) = x−t x−t 0 0

7.3 The Generalized Abel’s Integral Equation Abel generalized his original problem by introducing the singular integral equation x 1 f (x) = u(t)dt, 0 < α < 1, (7.35) (x − t)α 0

7.3 The Generalized Abel’s Integral Equation

243

known as the Generalized Abel’s integral equation where α are known constants such that 0 < α < 1, f (x) is a predetermined data function, and u(x) is the solution that will be determined. The Abel’s problem discussed above is a special case of the generalized equation where α = 12 . The expression (x − t)−α is called the kernel of the Abel integral equation, or simply Abel’s kernel.

7.3.1 The Laplace Transform Method To determine a formula that will be used for solving the generalized Abel’s integral equation (7.35), we will apply the Laplace transform method in a parallel manner to the approach followed before. Taking Laplace transforms of both sides of (7.35) leads to (7.36) L{f (x)} = L{u(x)}L{x−α }, or equivalently Γ(1 − α) F (s) = U (s) 1−α , (7.37) s that gives s1−α F (s), (7.38) U (s) = Γ(1 − α) where Γ is the gamma function. The last equation (7.38) can be rewritten as s L{u(x)} = L(y(x)), (7.39) Γ(α)Γ(1 − α) where x 1 y(x) = f (t)dt. (7.40) α−1 0 (x − t) Using the facts

L{y (x)} = sL{y(x)} − y(0),

and Γ(α)Γ(1 − α) =

π , sin απ

(7.41) (7.42)

into (7.39) we obtain sin απ L(y (x)). (7.43) π Applying L−1 to both sides of (7.43) gives the formula x f (t) sin απ d u(x) = dt. (7.44) π dx 0 (x − t)1−α Integrating the integral at the right side of (7.44) and diﬀerentiating the result we obtain the more suitable x formula f (t) sin απ f (0) + dt , 0 < α < 1. (7.45) u(x) = 1−α π x1−α 0 (x − t) L{u(x)} =

244

7 Abel’s Integral Equation and Singular Integral Equations

Four remarks can be made here: 1. The kernel is called weakly singular as the singularity may be transformed away by a change of variable [5]. 2. The exponent of the kernel of the generalized Abel’s integral equation is −α, but the exponent of the kernel of the two formulae (7.44) and (7.45) is α − 1. 3. The unknown function in (7.35) has been replaced by f (t) and f (t) in (7.44) and (7.45) respectively. 4. In (7.44), diﬀerentiation is used after integrating the integral at the right side, whereas in (7.45), integration is only required. Example 7.5 Solve the following generalized Abel’s integral equation x 128 114 1 x = 1 u(t)dt. 231 0 (x − t) 4

(7.46)

11

4 Notice that α = 14 , f (x) = 128 231 x . Using (7.44) gives √ x 128 11 t4 1 d 2 3 1 d 231 √ = x2 . u(x) = √ πx 3 dt = 3 2π dx 0 (x − t) 4 2π dx

(7.47)

Example 7.6 Solve the following generalized Abel’s integral equation x 1 πx = 2 u(t)dt. 0 (x − t) 3 Notice that α = 23 , f (x) = πx. Using (7.44) gives √ √ √ x 3 3 2 3 d πt 3 d 9 53 u(x) = dt = = x x3 . 2π dx 0 (x − t) 13 2 dx 10 4

(7.48)

(7.49)

Example 7.7 Solve the following generalized Abel’s integral equation x 1 36 11 x6 = 1 u(t)dt. 55 (x − t) 6 0 36 11 x 6 . Using 55 x 36 11 t6 d 55

Notice that α = 16 , f (x) = u(x) =

1 2π dx

0

(x − t)

5 6

(7.50)

(7.44) gives dt =

1 d (πx2 ) = x. 2π dx

(7.51)

Example 7.8 Solve the following generalized Abel’s integral equation x 512 15 1 4 x = 1 u(t)dt. 1155 0 (x − t) 4 Notice that α = 14 , f (x) =

512 15 4 1155 x .

Using (7.44) gives

(7.52)

7.3 The Generalized Abel’s Integral Equation

1 d u(x) = √ 2π dx

0

x

245

512 15 4 1155 t

3

(x − t) 4

dt = x3 .

(7.53)

7.3.2 The Main Generalized Abel Equation It is useful to introduce a further generalization to Abel’s equation by consid1 ering a generalized singular kernel instead of K(x, t) = √x−t . The generalized kernel will be of the form 1 , 0 < α < 1. (7.54) K(x, t) = [g(x) − g(t)]α The main generalized Abel’s integral equation is given by x 1 f (x) = u(t)dt, 0 < α < 1, (7.55) [g(x) − g(t)]α 0 where g(t) is strictly monotonically increasing and diﬀerentiable function in some interval 0 < t < b, and g (t) = 0 for every t in the interval. The solution u(x) of (7.55) is given by x g (t)f (t) sin απ d u(x) = dt, 0 < α < 1. (7.56) π dx 0 [g(x) − g(t)]1−α To prove this formula, we follow [6–7] and consider the integral x g (y)f (y) dy, 1−α 0 [g(x) − g(y)] and substitute for f (u) from (7.55) to obtain x y u(t)g (y) α 1−α dtdy, 0 0 [g(y) − g(t)] [g(x) − g(y)] where by changing the order of integration we ﬁnd y x g (y) u(t)dt α 1−α dy. 0 0 [g(y) − g(t)] [g(x) − g(y)] We can prove that y π g (y) , dy = β(α, 1 − α) = α 1−α sin απ 0 [g(y) − g(t)] [g(x) − g(y)] where β(α, 1 − α) is the beta function. This means that x x π g (y)f (y) dy = u(t)dt. 1−α sin απ 0 0 [g(x) − g(y)] Diﬀerentiating both sides of (7.58) gives x g (y)f (y) sin απ d dy, 0 < α < 1. u(x) = π dx 0 [g(x) − g(y)]1−α

(7.57)

(7.58)

(7.59)

246

7 Abel’s Integral Equation and Singular Integral Equations

It is interesting to point out that the Abel’s integral equation and the generalized Abel’s integral equation are special case of (7.55) by substituting g(x) = x respectively. The following illustrative examples explain how we can use the formula (7.56) in solving the further generalized Abel’s equation. Example 7.9 Solve the following generalized Abel’s integral equation x 4 3 u(t) 4 (sin x) = 1 dt, 3 0 (sin x − sin t) 4

(7.60)

3

where 0 < x < π2 . Notice that α = 14 , f (x) = 43 (sin x) 4 . Besides, g(x) = sin x is strictly monotonically increasing in 0 < x < π2 , and g (x) = cos x = 0 for every x in 0 < x < π2 . Using (7.56) gives 3 x 4 4 1 d 3 cos t(sin t) u(x) = √ 3 dt dx 2π 0 (sin x − sin t) 4 √ (7.61) 4 d 3 2π sin x = cos x. = √ 4 3 2π dx Example 7.10 Solve the following generalized Abel’s integral equation x 2 3 u(t) √ dt, (7.62) πx = 3 x2 − t2 0 where 0 < x < 2. Notice that α = 12 , f (x) = 23 πx3 . Besides, g(x) = x2 is strictly monotonically increasing in 0 < x < 2, and g (x) = 2x = 0 for every x in 0 < x < 2. Using (7.56) gives x 4 4 πt 1 d 1 d π2 4 √3 u(x) = dt = (7.63) x = πx3 . π dx 0 π dx 4 x2 − t 2 Example 7.11 Solve the following generalized Abel’s integral equation x u(t) √ x2 = dt, (7.64) 2 − t2 x 0 where 0 < x < 2. Notice that α = 12 , f (x) = x2 . Besides, g(x) = x2 is strictly monotonically increasing in 0 < x < 2, and g (x) = 2x = 0 for every x in 0 < x < 2. Using (7.56) gives x 1 d 4 3 2t3 4 1 d √ dt = (7.65) u(x) = x = x2 . 2 2 π dx 0 π dx 3 π x −t Example 7.12 Solve the following generalized Abel’s integral equation

7.4 The Weakly Singular Volterra Equations

6 25 x6 = 25

0

x

247

u(t) (x5

1

− t5 ) 6

dt,

(7.66)

25

6 x 6 . Besides, because g(x) is where 0 < x < 2. Notice that α = 16 , f (x) = 25 strictly monotonically increasing in 0 < x < 2, and g (x) = 5x4 = 0 for every x in 0 < x < 2. Using (7.56) gives x 6 49 t6 1 d 1 d 2 5 5 πx = x4 . u(x) = dt = (7.67) 2π dx 0 (x5 − t5 ) 56 2π dx 5

Exercises 7.3 In Exercises 1–8, solve the following generalized Abel’s integral equations x x 9 4 1 1 432 17 x6 = 1. x 3 = 2. 2 u(t)dt 1 u(t)dt 4 935 0 (x − t) 3 0 (x − t) 6 x x 1 1 3 2 9 5 24 5 x6 = x3 = 4. x 3 + 3. 1 u(t)dt 1 u(t)dt 5 2 10 6 0 (x − t) 0 (x − t) 3 x x 1 1 1 243 11 27 7 5. 6. 2πx 3 − x3 = x3 = 1 u(t)dt 2 u(t)dt 440 14 0 (x − t) 3 0 (x − t) 3 x x 1 1 128 11 25 9 16 7 5 = u(t)dt 8. 7. x4 + x4 = x 1 1 u(t)dt 21 231 36 0 (x − t) 4 0 (x − t) 5 In Exercises 9–16, use the formula (7.56) to solve the main generalized Abel’s integral equations x x 3 3 1 1 10. x2 = 9. 2x 2 = 1 u(t)dt 1 u(t)dt 2 0 (x3 − t3 ) 2 0 (x3 − t3 ) 3 x x 2 3 4 1 1 3 3 = 11. x 3 = (sin x) u(t)dt 12. 1 1 u(t)dt 4 2 0 (x2 − t2 ) 3 0 (sin x − sin t) 3 x x 5 1 1 6 2√ x 13. (ex − 1) 6 = 14. e − 1(2ex + 1) = 1 u(t)dt 1 u(t)dt 5 3 0 (ex − et ) 6 0 (ex − et ) 2 x x √ 1 1 64 11 x2 = 15. 2 sin x = 1 u(t)dt 16. 1 u(t)dt 2 231 2 0 (sin x − sin t) 0 (x − t2 ) 4

7.4 The Weakly Singular Volterra Equations The weakly-singular Volterra-type integral equations of the second kind are given by x β √ u(t)dt, x ∈ [0, T ], (7.68) u(x) = f (x) + x −t 0 and

248

7 Abel’s Integral Equation and Singular Integral Equations

x

β (7.69) α u(t)dt, 0 < α < 1, x ∈ [0, T ], 0 [g(x) − g(t)] where β is a constant. Equation. (7.69) is known as the generalized weakly singular Volterra equation. These equations arise in many mathematical physics and chemistry applications such as stereology, heat conduction, crystal growth and the radiation of heat from a semi-inﬁnite solid. It is also assumed that the function f (x) is suﬃciently smooth so that it guarantees a unique solution to (7.68) and to (7.69). The weakly-singular and the generalized weakly-singular equations (7.68) and (7.69) fall under the category of singular equations with singular kernels 1 K(x, t) = √ , x−t (7.70) 1 , K(x, t) = α [g(x) − g(t)] respectively. Notice that the kernel is called weakly singular as the singularity may be transformed away by a change of variable [5]. The weakly-singular Volterra equations (7.68) and (7.69) were investigated by many analytic and numerical methods. In this text we will use only three methods that were used before for handling Volterra integral equations in Chapter 3. u(x) = f (x) +

7.4.1 The Adomian Decomposition Method The Adomian decomposition method has been discussed extensively in this text. We will focus our study on the generalized weakly singular Volterra equation (7.69), because Eq. (7.68) is a special case of the generalized equation with α = 12 , g(x) = x. In the following we outline a brief framework of the method. To determine the solution u(x) of (7.69) we substitute the decomposition series ∞ un (x), (7.71) u(x) = n=0

into both sides of (7.69) to obtain ∞ x ∞ β un (x) = f (x) + un (t) dt, 0 < α < 1. α 0 [g(x) − g(t)] n=0 n=0

(7.72)

The components u0 (x), u1 (x), u2 (x), · · · are usually determined by using the recurrence relation u0 (x) = f (x) x β u1 (x) = α u0 (t)dt, 0 [g(x) − g(t)]

7.4 The Weakly Singular Volterra Equations

u2 (x) = u3 (x) =

x

0 x 0

β u1 (t)dt, [g(x) − g(t)]α β α u2 (t)dt, [g(x) − g(t)]

249

(7.73)

.. . Having determined the components u0 (x), u1 (x), u2 (x), . . ., the solution u(x) of (7.69) will be determined in the form of a rapid convergent power series [8] by substituting the derived components in (7.71). The determination of the previous components can be obtained by using Appendix B, calculator, or any computer symbolic systems such as Maple or Mathematica. The series solution converges to the exact solution if such a solution exists. For concrete problems, we use as many terms as we need for numerical purposes. The method has been proved to be eﬀective in handling this kind of integral equations.

The Modiﬁed Decomposition Method The modiﬁed decomposition method has been used eﬀectively in this text. Recall that this method splits the source term f (x) into two parts f1 (x) and f2 (x), where the ﬁrst part is f1 (x) assigned to the zeroth component u0 (x) and the other part f2 (x) to the ﬁrst component u1 (x). Based on this decomposition of f (x), we use the modiﬁed recurrence relation as follows: u0 (x) = f1 (x), x β u1 (x) = f2 (x) + α u0 (t)dt, [g(x) − g(t)] 0 x β u2 (x) = α u1 (t)dt, (7.74) [g(x) − g(t)] 0 x β u3 (x) = α u2 (t)dt, [g(x) − g(t)] 0 .. . It was proved before that the modiﬁed method accelerates the convergence of the solution.

The Noise Terms Phenomenon The noise terms that may appear between various components of u(x) are deﬁned as the identical terms with opposite signs. By canceling these noise terms between the components u0 (x) and u1 (x) may give the exact solution that should be justiﬁed through substitution. The noise terms, if appeared

250

7 Abel’s Integral Equation and Singular Integral Equations

between components of u(x), accelerate the convergence of the solution and thus minimize the size of the calculations. The noise terms will be used in the following examples. Example 7.13 Solve the weakly singular second kind Volterra integral equation x 16 5 1 2 2 √ u(x) = x + x − u(t)dt. 15 x−t 0 Using the recurrence relation we set 16 5 u0 (x) = x2 + x 2 , 15 x 2 16 5 t + 15 t 2 16 5 π √ dt = − x 2 − x3 . u1 (x) = − 15 3 x−t 0

(7.75)

(7.76)

5

2 The noise terms ± 16 15 x appear between u0 (x) and u1 (x). By canceling the 16 52 noise term 15 x from u0 (x) and verifying that the non-canceled term in u0 (x) justiﬁes the equation (7.75), the exact solution is therefore given by u(x) = x2 . (7.77)

Moreover, we can easily obtain the exact solution by using the modiﬁed decomposition method. This can be done by splitting the nonhomogeneous 5 2 part f (x) into two parts f1 (x) = x2 and f2 (x) = 16 15 x . Accordingly, we set the modiﬁed recurrence relation u0 (x) = x2 , x (7.78) 16 5 t2 2 √ x − u1 (x) = dt = 0. 15 x−t 0 Example 7.14 Solve the weakly singular second kind Volterra integral equation x 1 32 74 4 34 u(x) = 1 − 2x − x + x − (7.79) 1 u(t)dt. 21 3 0 (x − t) 4 Using the recurrence relation we set 32 7 4 3 u0 (x) = 1 − 2x − x 4 + x 4 , 21 3 7 3 x 1 − 2t − 32 t 4 + 43 t 4 21 (7.80) u1 (x) = − dt 1 (x − t) 4 0 4 3 128 5 16 3 32 7 x4 − x4 + x2 − x2 . = 21 3 63 9 3 32 74 x and ± 43 x 4 appear between u0 (x) and u1 (x). By The noise terms ∓ 21 canceling the noise terms from u0 (x) and verifying that the non-canceled term in u0 (x) justiﬁes the equation (7.79), the exact solution is therefore given by u(x) = 1 − 2x. (7.81)

7.4 The Weakly Singular Volterra Equations

251

Moreover, we can easily obtain the exact solution by using the modiﬁed decomposition method. This can be done by splitting the nonhomogeneous 3 32 74 part f (x) into two parts f1 (x) = 1 − 2x and f2 (x) = − 21 x + 43 x 4 . The use of the modiﬁed method is left as an exercise to the reader. Example 7.15 Solve the weakly singular second kind Volterra integral equation x 1 2 x x 3 u(t)dt. (7.82) u(x) = e − 3(e − 1) + x t 1 0 (e − e ) 3 In this example, the modiﬁed decomposition method will be used. We ﬁrst set 1 f1 (x) = ex , f2 (x) = −3(ex − 1) 3 . (7.83) In view of this, we use the modiﬁed recurrence relation x 1 2et x x 3 dt = 0. u0 (x) = e , u1 (x) = −3(e − 1) + x t 1 0 (e − e ) 3 The exact solution is therefore given by u(x) = ex .

(7.84)

(7.85)

Example 7.16 Solve the weakly singular second-type Volterra integral equation 1 1 1 x u(x) = sin x + (cos x − 1) 6 + u(t)dt. (7.86) 6 0 (cos x − cos t) 56 In this example, we will also use the modiﬁed decomposition method. We ﬁrst set 1 f1 (x) = sin x, f2 (x) = (cos x − 1) 6 . (7.87) In view of this, we use the modiﬁed recurrence relation u0 (x) = sin x, 1 x sin t 1 dt = 0. u1 (x) = (cos x − 1) 6 + 6 0 (cos x − cos t) 56 The exact solution is therefore given by u(x) = sin x.

(7.88)

(7.89)

Example 7.17 Solve the weakly singular second kind Volterra integral equation x 1 3 4 9 10 3 3 3 u(x) = x + x − x − x + u(t)dt. (7.90) 2 2 1 4 20 0 (x − t ) 3 In this example, we will also use the modiﬁed decomposition method. We ﬁrst set 3 4 9 10 (7.91) f1 (x) = x + x3 , f2 (x) = − x 3 − x 3 . 4 20

252

7 Abel’s Integral Equation and Singular Integral Equations

In view of this, we use the modiﬁed recurrence relation x 3 4 9 10 t + t3 u0 (x) = x + x3 , u1 (x) = − x 3 − x 3 + dt = 0. (7.92) 2 2 1 4 20 0 (x − t ) 3 The exact solution is therefore given by (7.93) u(x) = x + x3 . Example 7.18 As a ﬁnal example we consider the weakly singular second kind Volterra integral equation x √ 1 √ u(t)dt. (7.94) u(x) = 2 x − x−t 0 Following the discussion in the previous examples we use the recurrence relation √ u0 (x) = 2 x, x √ t √ dt = −πx, u1 (x) = −2 x − t 0 x πt 4 √ u2 (x) = dt = πx3/2 , (7.95) 3 x−t 0 4 x πt3/2 1 √ u3 (x) = − dt = − π 2 x2 , 3 0 2 x−t .. . Because self-canceling noise terms did not appear in the components u0 (x) and u1 (x), we obtained more components. Consequently, the solution u(x) in a series form is given by √ 4 1 (7.96) u(x) = 2 x − πx + πx3/2 − π 2 x2 + · · · , 3 2 and in a closed form by √ u(x) = 1 − eπx erfc( πx), (7.97) where erfc is the complementary error function normally used in probability topics. The deﬁnitions of the error function and the complementary error function can be found in Appendix D.

Exercises 7.4.1 Solve the following weakly singular Volterra equations, where x ∈ [0, T ]. x 9 4 1 1. u(x) = x − x 3 + 2 u(t)dt 4 0 (x − t) 3 x 1 9 5 x3 + 2. u(x) = x − 1 u(t)dt 10 0 (x − t) 3

7.4 The Weakly Singular Volterra Equations x 2 1 3 3. u(x) = sin x + (cos x − 1) 3 + 1 u(t)dt 2 0 (cos x − cos t) 3 x 2 1 2 4. u(x) = cos x − (sin x) 3 + 1 u(t)dt 3 0 (sin x − sin t) 3 x 2 1 3 5. u(x) = e−x + (e−x − 1) 3 + 1 u(t)dt 2 0 (e−x − e−t ) 3 x 1 3 4 6. u(x) = x − x 3 + 1 u(t)dt 2 4 0 (x − t2 ) 3 x 1 8 7 7. u(x) = x3 − x2 + 1 u(t)dt 2 21 0 (x − t2 ) 4 x 1 2 8. u(x) = x3 + 1 u(t)dt 4 3 0 (x − t4 ) 4 x 1 16 21 x4 + 9. u(x) = x5 − 1 u(t)dt 63 0 (x3 − t3 ) 4 x 1 7 5 x + 10. u(x) = 1 u(t)dt 10 0 (x3 − t3 ) 3 x 6 7 1 11. u(x) = ex − (ex − 1) 7 + 1 u(t)dt 6 0 (ex − et ) 7 x 2 1 3 x 12. u(x) = ex (ex + 1) − (e − 1) 3 (3ex + 7) + 1 u(t)dt 10 0 (ex − et ) 3 x 2 3 1 13. u(x) = sin 2x + (3 cos x + 2)(cos x − 1) 3 + 1 u(t)dt 5 0 (cos x − cos t) 3 x 2 3 1 14. u(x) = sinh x − (cosh x − 1) 3 + 1 u(t)dt 2 0 (cosh x − cosh t) 3 x 1 π 15. u(x) = 1 − + 1 u(t)dt 2 2 0 (x − t2 ) 2 x 1 π2 π 16. u(x) = π − + 1− x2 + 1 u(t)dt 2 2 4 0 (x − t2 ) 2

253

7.4.2 The Successive Approximations Method The successive approximations method, or Picard iteration method provides a scheme that can be used for solving initial value problem or integral equation. The method was used before in Chapter 3. This method solves any problem by ﬁnding successive approximations to the solution by starting with an initial guess, called the zeroth approximation. As will be seen, the zeroth approximation is any selective real-valued function that will be used in a recurrence relation to determine the other approximations.

254

7 Abel’s Integral Equation and Singular Integral Equations

In this section, we will study the weakly singular Volterra integral equation of the second kind x 1 √ u(x) = f (x) + u(t)dt. (7.98) x −t 0 The successive approximations method introduces the recurrence relation x 1 √ un (t)dt, n 0, un+1 (x) = f (x) + (7.99) x −t 0 where the zeroth approximation u0 (x) can be any selective real valued function. We always start with an initial guess for u0 (x), mostly we select 0, 1, x for u0 (x), and by using (7.99), several successive approximations uk , k 1 will be determined as x 1 √ u0 (t)dt, u1 (x) = f (x) + x−t 0 x 1 √ u2 (x) = f (x) + (7.100) u1 (t)dt, x −t 0 x 1 √ u3 (x) = f (x) + u2 (t)dt, x −t 0 and so on. As stated before in Chapter 3, the solution u(x) is given by u(x) = lim un+1 (x). n→∞

(7.101)

The successive approximations method will be explained by studying the following illustrative questions. Example 7.19 Solve the weakly singular integral equation by using the successive approximations method x √ 1 π √ u(x) = x + x − u(t)dt. (7.102) 2 x −t 0 For the zeroth approximation u0 (x), we can select (7.103) u0 (x) = 0. The method of successive approximations admits the use of the iteration formula x √ π 1 √ un (t)dt, n 0. (7.104) un+1 (x) = x + x − 2 x −t 0 Substituting (7.103) into (7.104) we obtain √ π u1 (x) = x + x, 2 √ π 2 3 π 2 x+ x u2 (x) = x+ x − , 2 2 3 (7.105) .. . √ π 2 3 2 3 π π2 2 x + x2 + x2 + x − ··· . x+ x − un+1 (x) = 2 2 3 3 4

7.4 The Weakly Singular Volterra Equations

The solution u(x) is given by u(x) = lim un+1 (x) = n→∞

255

√ x.

(7.106)

Example 7.20 Solve the weakly singular integral equation by using the successive approximations method x 4 3 1 √ u(t)dt. (7.107) u(x) = x − x 2 + 3 x −t 0 For the zeroth approximation u0 (x), we can select (7.108) u0 (x) = 0. The method of successive approximations admits the use of the iteration formula x 4 3 1 2 √ (7.109) un+1 (x) = x − x + un (t)dt, n 0. 3 x−t 0 Proceeding as before we set 4 3 u1 (x) = x − x 2 , 3 4 3 4 3 π 2 2 2 + x − x , u2 (x) = x − x 3 3 2 4 3 π 2 8π 5 4 3 π u3 (x) = x − x 2 + x 2 − x2 + x − x2 , 3 3 2 2 15

(7.110)

.. .

4 3 π 4 3 π 2 8π 5 x − x2 + x 2 − x2 + x − x2 + · · · . 3 3 2 2 15 The solution u(x) is given by un+1 (x) =

u(x) = lim un+1 (x) = x. n→∞

(7.111)

Example 7.21 Solve the weakly singular integral equation x √ 4 3 1 2 √ u(t)dt. u(x) = 1 + x − 2 x − x + 3 x−t 0 For the zeroth approximation u0 (x), we can select u0 (x) = 0. Proceeding as before, we use of the iteration formula x √ 4 3 1 √ un+1 (x) = 1 + x − 2 x − x 2 + un (t)dt, n 0. 3 x−t 0 This in turn gives

(7.112)

(7.113)

256

7 Abel’s Integral Equation and Singular Integral Equations

√ u1 (x) = (1 + x) − 2 x + √ u2 (x) = (1 + x) − 2 x +

4 3 x2 , 3 √ 4 3 4 3 π 2 2 + 2 x+ x − πx + x2 , x 3 3 2

.. .

π π un+1 (x) = (1 + x) − πx + x2 + πx + x2 + · · · . 2 2 The solution u(x) is given by u(x) = lim un+1 (x) = 1 + x. n→∞

(7.114) (7.115)

Example 7.22 We ﬁnally consider the weakly singular Volterra integral equation x √ 1 √ u(t)dt. u(x) = 2 x − x−t 0 We select u0 (x) = 0, hence we use the recurrence relation √ u1 (x) = 2 x, x √ √ u0 (t) √ u2 (x) = 2 x − dt = 2 x − πx, x−t 0 .. . √ 4 3 1 un+1 (x) = 2 x − πx + x 2 − π 2 x2 + · · · . 3 2 This in turn gives √ u(x) = 1 − eπx erfc( πx).

(7.116)

(7.117)

(7.118)

Exercises 7.4.2 Use the successive approximations method to solve the following weakly singular Volterra equations: x √ 1 √ 1. u(x) = x − 2πx + 4 u(t)dt x−t 0 x 5 1 5 √ πx3 + u(t)dt 2. u(x) = x 2 − 16 x−t 0 x √ 1 √ u(t)dt 3. u(x) = 3 − 6 x + x −t 0 x √ 1 1 √ u(t)dt 4. u(x) = 1 − x − πx + 2 x−t 0 x 3 1 3 √ u(t)dt 5. u(x) = x 2 − πx2 + 8 x−t 0 x 1 32 7 √ x2 + u(t)dt 6. u(x) = x3 − 33 x −t 0

7.4 The Weakly Singular Volterra Equations x √ 1 16 5 7. u(x) = 1 + x2 − 2 x − √ x2 + u(t)dt 15 x−t 0 x √ 1 4 3 √ u(t)dt 8. u(x) = 1 − x − 2 x + x 2 + 3 x −t 0 x 1 16 5 √ x2 + u(t)dt 9. u(x) = x2 − 15 x −t 0 x √ 4 1 π √ x + x(1 − x) + u(t)dt 10. u(x) = 1 − 2 3 x−t 0 x √ 1 √ u(t)dt 11. u(x) = π(1 − 2 x) + x −t 0 x 4 3 1 4 u(t)dt 12. u(x) = x(1 + x) − x 2 1 + x + √ 3 5 x−t 0

257

7.4.3 The Laplace Transform Method The Laplace transform method is used before, and the properties were presented in details in Chapters 1, 3, 5, and in the previous section. Recall that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by L{(f1 ∗ f2 )(x)} = F1 (s)F2 (s). (7.119) We will focus our study on the generalized weakly singular Volterra equation of the form x 1 u(t)dt, 0 < α < 1. (7.120) u(x) = f (x) + α 0 (x − t) Taking Laplace transforms of both sides of (7.120) leads to (7.121) L(u(x)) = L(f (x)) + L(x−α )L(u(x)), or equivalently Γ(1 − α) U (s) = F (s) + U (s), (7.122) s1−α that gives s1−α F (s) , (7.123) U (s) = 1−α s − Γ(1 − α) where Γ is the gamma function, U (s) = L{u(x)}, and F (s) = L{f (x)}. In Appendix D, the deﬁnition of the gamma function and some of the relations related to it are given. Applying L−1 to both sides of (7.123) gives the formula s1−α F (s) −1 , (7.124) u(x) = L s1−α − Γ(1 − α) that will be used for the determination of the solution u(x). Notice that the formula (7.124) will be used for solving the weakly singular integral equation, and it is not necessary to use Laplace transform method for each problem.

258

7 Abel’s Integral Equation and Singular Integral Equations

The following weakly singular Volterra equations will be examined by using the Laplace transform method. Example 7.23 Use the Laplace transform method to solve the weakly singular Volterra integral equation x √ 1 √ u(x) = 1 − 2 x + u(t)dt. (7.125) x−t 0 √ Notice that f (x) = 1 − 2 x and α = 12 . This means that √ Γ( 32 ) 1 π 1 (7.126) F (s) = − 2 3 = − 3 , s s s2 s2 √ where we used Γ( 32 ) = 12 Γ( 12 ) = 2π . Substituting (7.126) into the formula given in (7.124) gives √ ⎞ ⎛ 1 s 2 1s − 3π 1 −1 ⎝ −1 s2 ⎠ =L u(x) = L = 1. (7.127) √ 1 s 2 s − π Example 7.24 Use the Laplace transform method to solve the weakly singular Volterra integral equation x 1 4 3 √ u(x) = x − x 2 + u(t)dt. (7.128) 3 x −t 0 3

Notice that f (x) = x − 43 x 2 and α = 12 . This means that √ 1 4 Γ( 52 ) 1 π F (s) = 2 − = 2− 5 , (7.129) s 3 s 52 s 2 s √ where we used Γ( 52 ) = 34 Γ( 12 ) = 34 π. Substituting (7.129) into (7.124) gives ⎛ 1 ⎞ √ s 2 s12 − 5π

s2 ⎠ = L−1 12 = x. (7.130) u(x) = L−1 ⎝ 1 √ s s2 − π

Example 7.25 Use the Laplace transform method to solve the weakly singular Volterra integral equation x 1 9 5 u(x) = x − x 3 + (7.131) 1 u(t)dt. 10 0 (x − t) 3 Notice that f (x) = x −

9 53 x 10

and α = 13 . This means that

Γ( 23 ) 9 Γ( 83 ) 1 1 − = − , (7.132) 8 8 s2 10 s 3 s2 s3 2 where we used Γ( 83 ) = 10 9 Γ( 3 ). Substituting (7.132) into (7.124) gives F (s) =

7.4 The Weakly Singular Volterra Equations

⎛ u(x) = L−1 ⎝

2

s3

1 s2 2 s3

−

Γ( 2 ) 3 8 s3

−Γ( 23 )

259 ⎞

⎠ = L−1

1 s2

= x.

(7.133)

Exercises 7.4.3 In Exercises 1–8, use the Laplace transform method to solve the following weakly singular Volterra equations: x √ 1 √ u(t)dt 1. u(x) = x − 2πx + 4 x −t 0 x 1 16 5 √ x2 + u(t)dt 2. u(x) = x2 − 15 x −t 0 x √ 1 √ u(t)dt 3. u(x) = 4 − 8 x + x−t 0 x 1 32 7 √ x2 + u(t)dt 4. u(x) = x3 − 33 x−t 0 x √ 1 1 √ u(t)dt 5. u(x) = 1 − x − πx + 2 x −t 0 x √ 1 4 3 u(t)dt 6. u(x) = 1 + x − 2 x − x 2 + √ 3 x−t 0 x 3 1 3 u(t)dt 7. u(x) = x 2 − πx2 + √ 8 x−t 0 x √ π 1 4 u(t)dt 8. u(x) = (1 − )x + x(1 − x) + √ 2 3 x −t 0 In Exercises 9–16, use the Laplace transform method to solve the generalized weakly singular Volterra equations: x 9 4 1 9. u(x) = x − x 3 + 2 u(t)dt 4 0 (x − t) 3 x 27 8 1 x3 + 10. u(x) = x2 − 1 u(t)dt 40 0 (x − t) 3 x 1 16 7 11. u(x) = x − x4 + 1 u(t)dt 21 0 (x − t) 4 x 1 243 11 12. u(x) = x3 − x3 + 1 u(t)dt 440 0 (x − t) 3 x 1 9 5 3 2 x3 − x3 + 13. u(x) = 1 + x − 1 u(t)dt 10 2 0 (x − t) 4 x 36 11 1 x6 + 14. u(x) = x − 1 u(t)dt 55 0 (x − t) 6 x 1 25 7 x5 + 15. u(x) = x − 3 u(t)dt 14 0 (x − t) 5 x 1 128 9 16. u(x) = x2 − x4 + 3 u(t)dt 45 0 (x − t) 4

260

7 Abel’s Integral Equation and Singular Integral Equations

References 1. V. Singh, R. Pandey and O. Singh, New stable numerical solutions of singular integral equations of Abel type by using normalized Bernstein polynomials, Appl. Math. Sciences, 3 (2009) 241–255. 2. A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 3. M. Masujima, Applied Mathematical Methods in Theoretical Physics, Wiley, Weinheim, (2005). 4. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 5. R. Estrada and R. Kanwal, Singular Integral Equations, Birkhauser, Berlin, (2000). 6. R. P. Kanwal, Linear Integral Equations, Birkhauser, Boston, (1997). 7. I.N. Sneddon, Mixed boundary value problems in potential theory, Wiley, New York, (1966). 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).

Chapter 8

Volterra-Fredholm Integral Equations

8.1 Introduction The Volterra-Fredholm integral equations [1–2] arise from parabolic boundary value problems, from the mathematical modelling of the spatio-temporal development of an epidemic, and from various physical and biological models. The Volterra-Fredholm integral equations appear in the literature in two forms, namely x b K1 (x, t)u(t)dt + λ2 K2 (x, t)u(t)dt, (8.1) u(x) = f (x) + λ1 0

a

and the mixed form

x

b

u(x) = f (x) + λ

K(r, t)u(t)dtdr, 0

(8.2)

a

where f (x) and K(x, t) are analytic functions. It is interesting to note that (8.1) contains disjoint Volterra and Fredholm integrals, whereas (8.2) contains mixed Volterra and Fredholm integrals. Moreover, the unknown functions u(x) appears inside and outside the integral signs. This is a characteristic feature of a second kind integral equation. If the unknown functions appear only inside the integral signs, the resulting equations are of ﬁrst kind. Examples of the two types of the Volterra-Fredholm integral equations of the second kind are given by x 1 xu(t)dt − tu(t)dt, (8.3) u(x) = 6x + 3x2 + 2 − 0

and u(x) = x +

17 2 x − 2

0

0

x 0

1

(r − t)u(t)drdt.

(8.4)

However, the Volterra-Fredholm integro-diﬀerential equations that appear in scientiﬁc applications will be presented in the next chapter.

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

262

8 Volterra-Fredholm Integral Equations

8.2 The Volterra-Fredholm Integral Equations In this section, we will study some of the reliable methods that will be used for analytic treatment of the Volterra-Fredholm integral equations [1–2] of the form b x K1 (x, t)u(t)dt + K2 (x, t)u(t)dt. (8.5) u(x) = f (x) + 0

a

This type of equations will be handled by using the Taylor series method and the Adomian decomposition method combined with the noise terms phenomenon or the modiﬁed decomposition method. Other approaches exist in the literature but will not be presented in this text.

8.2.1 The Series Solution Method The series solution method was examined before in this text . A real function u(x) is called analytic if it has derivatives of all orders such that the generic form of Taylor series at x = 0 can be written as ∞ a n xn . (8.6) u(x) = n=0

In this section we will apply the series solution method [3–4], that stems mainly from the Taylor series for analytic functions, for solving VolterraFredholm integral equations. We will assume that the solution u(x) of the Volterra-Fredholm integral equation b x K1 (x, t)u(t)dt + K2 (x, t)u(t)dt, (8.7) u(x) = f (x) + 0

a

is analytic, and therefore possesses a Taylor series of the form given in (8.6), where the coeﬃcients an will be determined recurrently. In this method, we usually substitute the Taylor series (8.6) into both sides of (8.7) to obtain ∞ x ∞ k k ak x = T (f (x)) + K1 (x, t) ak t dt k=0

0

b

K2 (x, t)

+

∞

a

2

a0 + a1 x + a2 x + · · · = T (f (x)) + + a

ak t

k

(8.8)

dt,

k=0

or for simplicity we use

k=0

b

x 0

K1 (x, t) a0 + a1 t + a2 t2 + · · · dt

K2 (x, t) a0 + a1 t + a2 t2 + · · · dt,

(8.9)

8.2 The Volterra-Fredholm Integral Equations

263

where T (f (x)) is the Taylor series for f (x). The Volterra-Fredholm integral equation (8.7) will be converted to a regular integral in (8.8) or (8.9) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integrals in (8.8) or (8.9), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (8.6). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. Example 8.1 Solve the Volterra-Fredholm integral equation by using the series solution method x 1 u(x) = −5 − x + 12x2 − x3 − x4 + (x − t)u(t)dt + (x + t)u(t)dt. (8.10) 0

0

Substituting u(x) by the series u(x) =

∞

an xn ,

(8.11)

n=0

into both sides of Eq. (8.10) leads to ∞ an xn = −5 − x + 12x2 − x3 − x4 + n=0

+

1

(x + t)

0

∞

an t

n

x 0

(x − t)

∞

an t

n=0

n

dt (8.12)

dt.

n=0

Evaluating the integrals at the right side, using few terms from both sides, and collecting the coeﬃcients of like powers of x, we ﬁnd a0 + a1 x + a2 x2 + a3 x3 + · · · 1 1 1 1 1 = −5 + a0 + a1 + a2 + a3 + a4 2 3 4 5 6 1 1 1 1 + −1 + a0 + a1 + a2 + a3 + a4 x 2 3 4 5 1 1 + 12 + a0 x2 + −1 + a1 x3 2 6

264

8 Volterra-Fredholm Integral Equations

1 + −1 + a2 x4 + · · · . (8.13) 12 Equating the coeﬃcients of like powers of x in both sides of (8.13), and solving the resulting system of equations, we obtain a0 = 0, a1 = 6, (8.14) a2 = 12, a3 = a4 = a5 = · · · = 0. The exact solution is therefore given by u(x) = 6x + 12x2 .

(8.15)

Example 8.2 Use the series solution method to solve the Volterra-Fredholm integral equation x 1 2 3 5 tu(t)dt + (x + t)u(t)dt. (8.16) u(x) = 2 − x − x − 6x + x + 0

−1

Substituting u(x) by the series u(x) =

∞

an xn ,

(8.17)

n=0

into both sides of Eq. (8.16) leads to ∞ n 2 3 5 an x = 2 − x − x − 6x + x + n=0

1

+

(x + t)

−1

∞

a n tn

x

0

dt.

t

∞

an tn dt

n=0

(8.18)

n=0

Evaluating the integrals at the right side, using few terms from both sides, and collecting the coeﬃcients of like powers of x, we ﬁnd a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · 2 2 = 2 + a1 + a3 3 5 2 2 + −1 + 2a0 + a2 + a4 x 3 5 1 1 2 + −1 + a0 x + −6 + a1 x3 2 3 1 1 + a2 x4 + 1 + a3 x5 + · · · . (8.19) 4 5 Equating the coeﬃcients of like powers of x in both sides of (8.19), and solving the resulting system of equations, we obtain a0 = 2, a1 = 3, a2 = 0, a3 = −5, (8.20) a4 = a5 = · · · = 0. The exact solution is therefore given by

8.2 The Volterra-Fredholm Integral Equations

265

u(x) = 2 + 3x − 5x3 .

(8.21)

Example 8.3 Solve the Volterra-Fredholm integral equation by using the series solution method x 1 x u(x) = e − 1 − x + u(t)dt + xu(t)dt. (8.22) 0

0

7

x

Using the Taylor polynomial for e up to x , substituting u(x) by the Taylor polynomial 7 u(x) = an xn , (8.23) n=0

and proceeding as before leads to 7 7 1 n an x an x = 2a0 + n+1 n=0 n=1 +

1 + a1 2 1 + 2!a2 3 1 + 3!a3 4 x + x + x 2! 3! 4!

1 + 4!a4 5 1 + 5!a5 6 1 + 6!a6 7 x + x + x + · · · . (8.24) 5! 6! 7! Equating the coeﬃcients of like powers of x in both sides of (8.19), and proceeding as before, we obtain 1 a0 = 0, a1 = 1, a2 = 1, a3 = , 2! (8.25) 1 1 1 a4 = , a5 = , a6 = . 3! 4! 5! The exact solution is given by (8.26) u(x) = xex . +

Example 8.4 Solve the Volterra-Fredholm integral equation by using the series solution method π x (x − t)u(t)dt + u(t)dt. (8.27) u(x) = 1 − 0

0

Substituting u(x) by the Taylor polynomial 8 u(x) = an xn ,

(8.28)

n=0

and proceeding as before, we obtain a0 = 1, a2 = −

a1 = a3 = a5 = a7 = 0, 1 , 21

a4 =

1 , 4!

(8.29)

266

8 Volterra-Fredholm Integral Equations

1 1 , a8 = . 6! 8! Consequently, the exact solution is given by u(x) = cos x. a6 = −

(8.30)

Exercises 8.2.1 Use the series solution method to solve the following Volterra-Fredholm integral equations x 1 (x − t)u(t)dt + (x + t)u(t)dt 1. u(x) = −4 − 3x + 11x2 − x3 − x4 + 2

0

3

2. u(x) = −1 − 3x − 2x +

x

(x − t)u(t)dt +

0

3. u(x) = 4 − x − 4x2 − x3 +

x 0

x 0

8. u(x) = −2x + x cos x + 9. u(x) = 2x −

x

0

0

x 0

0

π 2

12. u(x) = −2 − 2x + 2ex +

0

(x + t + 1)u(t)dt

0 1

(x + t)u(t)dt

0

1 0

(4xt2 + 3x2 t)u(t)dt

0

tu(t)dt

π 0

xu(t)dt

x

1

1

(x − t)u(t)dt −

u(t)dt +

(x + t − 1)u(t)dt

xu(t)dt

x 0

1

0

(x − t)u(t)dt +

xu(t)dt +

(x − t)u(t)dt −

11. u(x) = −x ln(1 + x) +

x

tu(t)dt +

(x − t)u(t)dt +

x

10. u(x) = 2 + x − 2 cos x −

(x + t)u(t)dt

x 0

0

0

(3xt + 4x2 t2 )u(t)dt +

7. u(x) = ex − xex + x − 1 +

x

0

5. u(x) = −6 − 2x + 19x3 − x5 + 6. u(x) = −x2 − x4 − x6 +

1

(x − t + 1)u(t)dt +

4. u(x) = −8 − 6x + 11x2 − x4 +

0

π 2

0

xu(t)dt

e−1

0

xu(t)dt

(x − t)u(t)dt +

1

0

xu(t)dt

8.2.2 The Adomian Decomposition Method The Adomian decomposition method [5–8] (ADM) was introduced thoroughly in this text for handling independently Volterra and Fredholm integral equations. The method consists of decomposing the unknown function u(x) of any equation into a sum of an inﬁnite number of components deﬁned by the decomposition series

8.2 The Volterra-Fredholm Integral Equations

u(x) =

∞

267

un (x),

(8.31)

n=0

where the components un (x), n 0 are to be determined in a recursive manner. To establish the recurrence relation, we substitute the decomposition series into the Volterra-Fredholm integral equation (8.5) to obtain ∞ ∞ x b ∞ un (x) = f (x) + K1 (x, t) un (t) dt + K2 (x, t) un (t) dt. 0

n=0

a

n=0

n=0

(8.32) The zeroth component u0 (x) is identiﬁed by all terms that are not included under the integral sign. Consequently, we set the recurrence relation u0 (x) = f (x), x b (8.33) K1 (x, t)un (t)dt + K2 (x, t)un (t)dt, n 0. un+1 (x) = 0

a

Having determined the components u0 (x), u1 (x), u2 (x), . . ., the solution in a series form is readily obtained upon using (8.31). The series solution converges to the exact solution if such a solution exists. We point here that the noise terms phenomenon and the modiﬁed decomposition method will be employed in this section. This will be illustrated by using the following examples. Example 8.5 Use the Adomian decomposition method to solve the following VolterraFredholm integral equation x 1 x u(x) = e + 1 + x + (x − t)u(t)dt − ex−t u(t)dt. (8.34) 0

0

Using the decomposition series (8.31), and using the recurrence relation (8.33) we obtain u0 (x) = ex + 1 + x, x 1 u1 (x) = (x − t)u0 (t)dt + ex−t u0 (t)dt (8.35) 0

0

1 = −x − 1 + x2 + · · · , 2 and so on. We notice the appearance of the noise terms ±1 and ±x between the components u0 (x) and u1 (x). By canceling these noise terms from u0 (x), the non-canceled term of u0 (x) gives the exact solution u(x) = ex , (8.36) that satisﬁes the given equation. It is to be noted that the modiﬁed decomposition method can be applied here. Using the modiﬁed recurrence relation u0 (x) = ex , x 1 (8.37) u1 (x) = 1 + x + (x − t)u0 (t)dt − ex−tu0 (t)dt = 0. 0

0

268

8 Volterra-Fredholm Integral Equations

The exact solution u(x) = ex follows immediately. Example 8.6 Use the modiﬁed Adomian decomposition method to solve the following Volterra-Fredholm integral equation x 1 1 1 1 u(x) = x2 − x4 − − x + (x − t)u(t)dt + (x + t)u(t)dt. (8.38) 12 4 3 0 0 Using the modiﬁed decomposition method gives the recurrence relation 1 u0 (x) = x2 − x4 , 12 1 x 1 1 (8.39) (x − t)u(t)dt + (x + t)u(t)dt u1 (x) = − − x + 4 3 0 0 1 1 4 1 6 1 x − x − x− . = 12 360 60 72 1 4 and so on. We notice the appearance of the noise terms ± 12 x between the components u0 (x) and u1 (x). By canceling the noise term from the u0 (x), the non-canceled term gives the exact solution u(x) = x2 , (8.40) that satisﬁes the given equation. Example 8.7 Use the modiﬁed Adomian decomposition method to solve the following Volterra-Fredholm integral equation π x u(t)dt + (x − t)u(t)dt. (8.41) u(x) = cos x − sin x − 2 + 0

0

Using the modiﬁed decomposition method gives the recurrence relation u0 (x) = cos x, x π (8.42) u(t)dt + (x − t)u(t)dt = 0. u1 (x) = − sin x − 2 + 0

0

Consequently, the exact solution is given by u(x) = cos x.

(8.43)

Example 8.8 Use the modiﬁed Adomian decomposition method to solve the following Volterra-Fredholm integral equation x 1 2 3 4 tu(t)dt + tu(t)dt. (8.44) u(x) = 3x + 4x − x − x − 2 + 0

−1

Using the modiﬁed decomposition method gives the recurrence relation u0 (x) = 3x + 4x2 − x3 , x 4 u1 (x) = −x − 2 + tu(t)dt + 0

1

−1

tu(t)dt

(8.45)

8.3 The Mixed Volterra-Fredholm Integral Equations

269

2 1 = − − x5 + x3 . 5 5 Canceling the noise term −x3 from u0 (x) gives the exact solution u(x) = 3x + 4x2 .

(8.46)

Exercises 8.2.2 Use the modiﬁed decomposition method to solve the following Volterra-Fredholm integral equations x 1 1 1. u(x) = x − x3 + tu(t)dt + t2 u(t)dt 3 0 −1 x π 4 u(t)dt + u(t)dt 2. u(x) = sec2 x − tan x − 1 + 3. u(x) = sin x − cos x − 4. u(x) = x3 −

0

x 0

u(t)dt +

1 5 1 1 x − x− + 20 4 5

x

0

0

π 2

0

u(t)dt

(x − t)u(t)dt +

1

(x + t)u(t)dt

0

x 1 9 5 1 1 (x + t)u(t)dt + (x − t)u(t)dt x − x+ + 20 4 5 0 0 x π 2 6. u(x) = sin x + cos x − x + u(t)dt + (x − t)u(t)dt

5. u(x) = x3 −

7. u(x) = cos x − sin x − 2 +

0 x

u(t)dt +

0

8. u(x) = cos x + (1 − x) sin x − 2 + 9. u(x) = tan x − ln(cos x) + 10. u(x) = cot x − ln(sin x) −

x

0

(x − t)u(t)dt

0

(x − 1)u(t)dt +

x 0

0 π

u(t)dt +

1 x ln(2) + 2

π 4

−π

0

(x − t)u(t)dt

xu(t)dt

4

x π 2

π

u(t)dt +

π 2 π 4

xu(t)dt

x π 1 1 2 1 x − + u(t)dt u(t)dt + π 2 2 0 4 2 x π 1 1 12. u(x) = x + cos x − sin x − x2 + 1 + u(t)dt u(t)dt + π 2 0 4 11. u(x) = x + sin x + cos x −

2

8.3 The Mixed Volterra-Fredholm Integral Equations In a parallel manner to our analysis in the previous section, we will study the mixed Volterra-Fredholm integral equations [2], given by

270

8 Volterra-Fredholm Integral Equations

x

b

u(x) = f (x) +

K(r, t)u(t)dtdr, 0

(8.47)

a

where f (x) and K(x, t) are analytic functions. It is interesting to note that (8.47) contains mixed Volterra and Fredholm integral equations. The Fredholm integral is the interior integral, whereas the Volterra integral is the exterior one. Moreover, the unknown function u(x) appears inside and outside the integral signs. This is a characteristic feature of a second kind integral equation. This type of equations will be handled by using the series solution method and the Adomian decomposition method, including the use of the noise terms phenomenon or the modiﬁed decomposition method. Other approaches exist in the literature but will not be presented in this text.

8.3.1 The Series Solution Method The series solution method was examined before in this chapter. A real function u(x) is called analytic if it has derivatives of all orders such that the generic form of Taylor series at x = 0 can be written as ∞ an xn . (8.48) u(x) = n=0

We will assume that the solution u(x) of the mixed Volterra-Fredholm integral equation x b K(r, t)u(t)dtdr, (8.49) u(x) = f (x) + 0

a

is analytic, and therefore possesses a Taylor series of the form given in (8.48), where the coeﬃcients an will be determined in an algebraic manner. In this method, we usually substitute the Taylor series (8.48) into both sides of (8.49) to obtain ∞ x b ∞ k k ak x = T (f (x)) + K(r, t) ak t dtdr, (8.50) 0

k=0

or for simplicity we use 2

a0 + a1 x + a2 x + · · · = T (f (x)) +

a

k=0

x b 0

a

K(r, t) a0 + a1 t + a2 t2 + · · · dtdr,

(8.51) where T (f (x)) is the Taylor series for f (x). The mixed Volterra-Fredholm integral equation (8.49) will be converted to a traditional integral in (8.50) or (8.51) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used.

8.3 The Mixed Volterra-Fredholm Integral Equations

271

We ﬁrst integrate the inner integral, and then we integrate the outer integral in (8.50) or (8.51), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine a system of algebraic equations in aj , j 0. Solving this system of equations will lead to a complete determination of the coefﬁcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (8.48). The exact solution may be obtained if such an exact solution exists, otherwise a truncated series is used for numerical approximations. Example 8.9 Solve the mixed Volterra-Fredholm integral equation by using the Taylor series solution method x 1 17 (r − t)u(t)dtdr. (8.52) u(x) = 11x + x2 + 2 0 0 Substituting u(x) by the series ∞ u(x) = an xn , (8.53) n=0

into both sides of Eq. (8.52) leads to x 1 ∞ ∞ 17 2 n n (r − t) dtdr. an x = 11x + x + an t 2 0 0 n=0 n=0

(8.54)

Evaluating the integrals at the right side, using few terms from both sides, collecting the coeﬃcients of like powers of x, and equating the coeﬃcients of like powers of x in both sides, we obtain a0 = 0, a1 = 6, (8.55) a2 = 12, a3 = a4 = a5 = · · · = 0. The exact solution is therefore given by u(x) = 6x + 12x2 .

(8.56)

Example 8.10 Solve the mixed Volterra-Fredholm integral equation by using the series solution method x 1 9 (r − t)u(t)dtdr. (8.57) u(x) = 2 + 4x − x2 − 5x3 + 8 0 0 Substituting u(x) by the series ∞ u(x) = an xn , (8.58) n=0

into both sides of Eq. (8.57) leads to x 1 ∞ ∞ 9 2 n 3 n (r − t) dt dr. (8.59) an x = 2 + 4x − x − 5x + an t 8 0 0 n=0 n=0

272

8 Volterra-Fredholm Integral Equations

Evaluating the integrals at the right side, using few terms from both sides, collecting the coeﬃcients of like powers of x, and equating the coeﬃcients of like powers of x in both sides, we obtain a0 = 2, a1 = 3, a2 = 0, a3 = −5, (8.60) a4 = a5 = · · · = 0. The exact solution is therefore given by (8.61) u(x) = 2 + 3x − 5x3 . Example 8.11 Solve the mixed Volterra-Fredholm integral equation by using the series solution method x 1 1 ru(t)dtdr. (8.62) u(x) = xex − x2 + 2 0 0 Using the Taylor polynomial for xex up to x9 , substituting u(x) by the Taylor polynomial 9 an xn , (8.63) u(x) = n=0

and proceeding as before, we obtain a0 = 0, a1 = 1, a2 = 1, 1 1 a4 = , a3 = , 2! 3! .. . The exact solution is therefore given by u(x) = xex .

a5 =

1 , 4!

(8.64)

(8.65)

Example 8.12 Solve the mixed Volterra-Fredholm integral equation by using the series solution method x π2 π u(x) = cos x + sin x − x2 + x + (r − t)u(t)dt dr. (8.66) 2 0 0 Proceeding as before we ﬁnd 1 a1 = 1, a2 = − , a0 = 1, 2! 1 1 1 (8.67) a 3 = − , a4 = , a5 = , 3! 4! 5! 1 1 1 a6 = − , a7 = − , a8 = 6! 7! 8! and so on. The series solution is given by 1 1 1 1 (8.68) u(x) = 1 − x2 + x4 + · · · + x − + x5 + · · · , 2! 4! 3! 5!

8.3 The Mixed Volterra-Fredholm Integral Equations

273

that converge to the exact solution u(x) = cos x + sin x.

(8.69)

Exercises 8.3.1 Use the series solution method to solve the following mixed Volterra-Fredholm integral equations x 1 15 2 x + 1. u(x) = 2 + 12x + (r − t)u(t)dtdr 2 0 0 x 1 (r − t)u(t)dtdr 2. u(x) = 6 + 19x − 6x2 + 3. u(x) = 2 + 10x − 2x2 + 4. u(x) = 2 + 6x2 + 5. u(x) = 7x +

x

0

31 2 x + 3

0

−1 x

6. u(x) = 4 + 14x − 2x2 +

0 1

0

0

x

−1

(r − t)u(t)dtdr

(r − t)u(t)dtdr

1 0

1

x

(r 2 − t2 )u(t)dtdr

0

1

−1

7. u(x) = 6 + 9x + 2x2 − 2x3 +

(r 3 − t3 )u(t)dtdr

x 0

1

−1

(rt2 + r2 t)u(t)dtdr

x π 1 2 2 x + (r − t)u(t)dtdr 2 0 0 x e−1 u(t)dtdr 9. u(x) = ln(1 + x) − x + 8. u(x) = sin x + x −

0

0

x e−1 1 10. u(x) = ln(1 + x) − x2 + ru(t)dtdr 2 0 0 x π 2 11. u(x) = cos x + 2 u(t)dtdr π 0 0 x π 1 u(t)dtdr 12. u(x) = tan x − x + ln 2 0 0

8.3.2 The Adomian Decomposition Method The Adomian decomposition method (ADM) was introduced thoroughly in this text for handling independently Volterra integral equations, Fredholm integral equations, and the Volterra-Fredholm integral equations. The method consists of decomposing the unknown function u(x) of any equation into a sum of an inﬁnite number of components deﬁned by the decomposition series

274

8 Volterra-Fredholm Integral Equations

u(x) =

∞

un (x).

(8.70)

n=0

To establish the recurrence relation, we substitute decomposition series into the mixed Volterra-Fredholm integral equation (8.49) to obtain ∞ x b ∞ un (x) = f (x) + K(r, t) un (t) dt dr. (8.71) 0

n=0

a

n=0

Consequently, we set the recurrence relation u0 (x) = f (x), x b un+1 (x) = K(r, t)un (t)dtdr, 0

n 0.

(8.72)

a

The solution in a series form is readily obtained upon using (8.70). The series may converge to the exact solution if such a solution exists. We point out here that the noise terms phenomenon and the modiﬁed decomposition method will be employed heavily. Example 8.13 Solve the mixed Volterra-Fredholm integral equation by using the Adomian decomposition method x 1 1 2 x ru(t)dtdr. (8.73) u(x) = xe − x + 2 0 0 Using the Adomian decomposition method we set the recurrence relation 1 u0 (x) = xex − x2 , 2 x 1 (8.74) ruk (t)dtdr, k 0. uk+1 (x) = 0

0

This in turn gives 1 u0 (x) = xex − x2 , 2 x 1 5 2 u1 (x) = x , ru0 (t)dtdr = 12 0 0 x 1 5 2 ru1 (t)dtdr = x , u2 (x) = 72 0 0 x 1 5 2 ru2 (t)dtdr = u3 (x) = x , 432 0 0 and so on. Using (8.70) gives the series solution 1 2 5 2 1 1 x u(x) = xe − x + x 1 + + + ··· , 2 12 6 36 that converges to the exact solution u(x) = xex ,

(8.75)

(8.76)

(8.77)

8.3 The Mixed Volterra-Fredholm Integral Equations

275

obtained upon ﬁnding the sum of the inﬁnite geometric series. However, we can obtain the exact solution by using the modiﬁed decomposition method. This can be achieved by setting u0 (x) = xex , x 1 (8.78) 1 u1 (x) = − x2 + ru0 (t)dtdr = 0. 2 0 0 Accordingly the other components uj (x), j 2 = 0. This gives the exact solution by u(x) = xex . (8.79) Example 8.14 Solve the mixed Volterra-Fredholm integral equation by using the Adomian decomposition method x π2 1 2 ru(t)dtdr. (8.80) u(x) = sin x − x + 2 0 0 Using the Adomian decomposition method we set the recurrence relation 1 u0 (x) = sin x − x2 , 2 x π2 (8.81) uk+1 (x) = ruk (t)dtdr, k 0. 0

0

This in turn gives 1 u0 (x) = sin x − x2 , 2 x π2 (8.82) 1 π3 2 u1 (x) = x . ru0 (t)dtdr = sin x + x2 − 2 96 0 0 1 2 By canceling the noise term − 2 x from u0 , the exact solution is given by u(x) = sin x. (8.83) This result satisﬁes the given equation. However, we can obtain the exact solution by using the modiﬁed decomposition method. This can be achieved by setting u0 (x) = sin x, x π2 (8.84) 1 ru0 (t)dtdr = 0. u1 (x) = − x2 + 2 0 0 Accordingly the other components uj (x), j 2 = 0. This gives the same solution obtained above. Example 8.15 Solve the mixed Volterra-Fredholm integral equation by using the Adomian decomposition method x π u(x) = cos x − 2x + (r − t)u(t)dtdr. (8.85) 0

0

276

8 Volterra-Fredholm Integral Equations

Using the modiﬁed decomposition method we set u0 (x) = cos x, x π (r − t)u0 (t)dtdr = 0. u1 (x) = −2x + 0

(8.86)

0

Accordingly the other components uj (x), j 2 = 0. This gives the exact solution by u(x) = cos x. (8.87) Example 8.16 Solve the mixed Volterra-Fredholm integral equation by using the Adomian decomposition method x e−1 1 ru(t)dtdr. (8.88) u(x) = ln(1 + x) − x2 + 2 0 0 Using the modiﬁed decomposition method we set u0 (x) = ln(1 + x), x e−1 (8.89) 1 u1 (x) = − x2 + ru0 (t)dtdr = 0. 2 0 0 Accordingly the other components uj (x), j 2 = 0. This gives the exact solution by u(x) = ln(1 + x). (8.90)

Exercises 8.3.2 In Exercises 1–6, use the Adomian decomposition method to solve the mixed VolterraFredholm integral equations x 1 7 1. u(x) = x + (r 3 − t3 )u(t)dtdr 5 0 −1 x 1 2 2. u(x) = x + x3 + (rt2 − r2 t)u(t)dtdr 9 0 −1 x 1 4 3. u(x) = x2 + (rt2 − r2 t)u(t)dtdr 5 0 −1 x 1 2 3 4. u(x) = (rt2 + r2 t)u(t)dtdr x + 15 0 −1 x 1 1 5. u(x) = x3 − x2 + (1 + rt)u(t)dtdr 5 0 −1 x 1 2 6. u(x) = x2 − x + (1 + rt)u(t)dtdr 3 0 −1 In Exercises 7–12, use the modiﬁed decomposition method to solve the following Volterra-Fredholm integral equations x π 1 2 (r − t)u(t)dtdr 7. u(x) = x sin x − x2 + (π − 2)x + 2 0 0

8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables π2 2 x + 4 9. u(x) = ln(1 + x) − x +

8. u(x) = x cos x −

x

0

π

ru(t)dtdr

0

x

0

10. u(x) = tan x − x2 ln 2 +

e−1

u(t)dtdr

0 x

0

277

π 3

0

ru(t)dtdr

x π π 1 4 (r + t)u(t)dtdr − ln 2)x − x2 + 4 2 0 0 x 2 ru(t)dtdr 12. u(x) = 2xex − (e2 + 1)x2 +

11. u(x) = sec2 x − (

0

0

8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables In this section we present a reliable strategy for solving the linear mixed Volterra-Fredholm integral equation of the form t F (x, t, r, s)u(r, s)drds, (x, t) ∈ Ω × [0, T ], (8.91) u(x, t) = f (x, t) + 0

Ω

where u(x, t) is an unknown function, and f (x, t) and F (x, t, r, s) are analytic functions on D = Ω × [0, T ], where Ω is a closed subset of Rn , n = 1, 2, 3. The mixed Volterra-Fredholm integral equation (8.91) in two variables arises from parabolic boundary value problems, the mathematical modeling of the spatio-temporal development of an epidemic, and in various physical and biological models. In the literature, some methods, such as the projection method, time collocation method, the trapezoidal Nystrom method, and Adomian method were used to handle this problem. It was found that these techniques encountered diﬃculties in terms of computational work used, and therefore, approximate solutions were obtained for numerical purposes. In particular, it was found that a complicated term f (x, t) can cause diﬃcult integrations and proliferation of terms in Adomian recursive scheme. To overcome the tedious work of the existing strategies, the modiﬁed decomposition method will form a useful basis for studying the mixed VolterraFredholm integral equation (8.91). The size of the computational work can be dramatically reduced by using the modiﬁed decomposition method. Moreover, the convergence can be accelerated by combining the modiﬁed method with the noise terms phenomenon. The noise terms were deﬁned as the identical terms with opposite signs that may appear in the ﬁrst two components of the series solution of u(x). The modiﬁed decomposition method and the noise terms phenomenon were presented in details in Chapters 3, 4, and 5.

278

8 Volterra-Fredholm Integral Equations

8.4.1 The Modiﬁed Decomposition Method In what follows we give a brief review of the modiﬁed decomposition method. For many cases, the function f (x, t) can be set as the sum of two partial functions, namely f1 (x, t) and f2 (x, t). In other words, we can set (8.92) f (x, t) = f1 (x, t) + f2 (x, t). To minimize the size of calculations, we identify the zeroth component u0 (x, t) by one part of f (x, t), namely f1 (x, t) or f2 (x, t). The other part of f (x, t) can be added to the component u1 (x, t) among other terms. In other words, the modiﬁed decomposition method introduces the modiﬁed recurrence relation u0 (x, t) = f1 (x, t), t u1 (x, t) = f2 (x, t) + F (x, t, r, s)u0 (r, s)drds, (8.93) 0 Ω t F (x, t, r, s)uk (r, s)drds, k 1. uk+1 (x, t) = Ω

0

If noise terms appear between u0 (x, t) and u1 (x, t), then by canceling these terms from u0 (x, t), the remaining non-canceled terms of u0 (x, t) may give the exact solution. This can be satisﬁed by direct substitution. In what follows, we study some illustrative examples. Example 8.17 Use the modiﬁed decomposition method to solve the mixed Volterra-Fredholm integral equation t u(x, t) = f (x, t) + F (x, t, r, s)u(r, s)drds, (x, t) ∈ Ω × [0, T ], (8.94) Ω

0

where

1 f (x, t) = e cos(x) + t cos(x) + t cos(x − 1) sin 1 , 2 F (x, t, r, s) = − cos(x − r)es−t , −t

with Ω = [0, 1]. Substituting the decomposition series ∞ u(x, t) = un (x, t),

(8.95)

(8.96)

n=0

into (8.94) gives ∞ 1 −t un (x, t) = e cos(x) + t cos(x) + t cos(x − 1) sin 1 2 n=0 ∞ t s−t − cos(x − r)e un (r, s) drds. 0

Ω

n=0

As stated before we decompose f (x, t) into two parts as follows:

(8.97)

8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables

279

f0 (x, t) = e−t [cos(x) + t cos(x)], (8.98) 1 −t te cos(x − 1) sin 1. 2 The modiﬁed decomposition technique admits the use of the recursive relation u0 (x, t) = e−t [cos(x) + t cos(x)], 1 u1 (x, t) = te−t cos(x − 1) sin 1 2 t (8.99) − cos(x − r)es−t u0 (r, s)drds f1 (x, t) =

0

t uk+1 (x, t) = −

0

Ω

Ω

cos(x − r)es−t uk (r, s)drds,

k 1.

This gives u0 (x, t) = e−t [cos(x) + t cos(x)], (8.100) 1 1 sin(x + 4) + sin2 2 sin x + · · · u1 (x, t) = −te−t cos(x) + te−t 2 2 −t The self-canceling noise terms te cos(x) and −te−t cos(x) appear between the components u0 (x, t) and u1 (x, t) respectively. By canceling this term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the equation (8.94), the exact solution is given by u(x, t) = e−t cos(x).

(8.101)

Example 8.18 Use the modiﬁed decomposition method to solve the mixed Volterra-Fredholm integral equation t F (x, t, r, s)u(r, s) dr ds, (x, t) ∈ Ω × [0, T ], (8.102) u(x, t) = f (x, t) + 0

where

Ω

π cos x(t cos t − sin t), 4 F (x, t, r, s) = cos(x − r) sin(t − s),

f (x, t) = cos x sin t +

with Ω = [0, π]. Substituting the decomposition series ∞ un (x, t), u(x, t) =

(8.103)

(8.104)

n=0

into (8.102) gives ∞ π un (x, t) = cos x sin t + cos x(t cos t − sin t) 4 n=0 ∞ t cos(x − r) sin(t − s) un (r, s) drds. + 0

Ω

n=0

(8.105)

280

8 Volterra-Fredholm Integral Equations

As stated before we decompose f (x, t) into two parts as follows: π f0 (x, t) = cos x sin t + t cos x cos t, 4 (8.106) π f1 (x, t) = − cos x sin t. 4 The modiﬁed decomposition technique admits the use of the recursive relation π u0 (x, t) = cos x sin t + t cos x cos t, 4 t π u1 (x, t) = − cos x sin t + cos(x − r) sin (t − s)u0 (r, s)drds (8.107) 4 0 Ω t cos(x − r) sin (t − s)uk (r, s)drds, k 1. uk+1 (x, t) = 0

Ω

This gives u0 (x, t) = cos x sin t +

π t cos x cos t, 4

(8.108) π u1 (x, t) = − t cos x cos t + · · · . 4 π The self-canceling noise terms 4 t cos x cos t and − π4 t cos x cos t appear between the components u0 (x, t) and u1 (x, t) respectively. By canceling this term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the equation (8.102), the exact solution u(x, t) = cos x sin t, (8.109) follows immediately. Example 8.19 Use the modiﬁed decomposition method to solve the mixed Volterra-Fredholm integral equation t F (x, t, r, s)u(r, s) dr ds, (x, t) ∈ Ω × [0, T ], (8.110) u(x, t) = f (x, t) + 0

Ω

where f (x, t) = sin(x + t) − 4 cos t − π sin t − 2t sin t + 4, F (x, t, r, s) = r + s, with Ω = [0, π]. Substituting the decomposition series ∞ u(x, t) = un (x, t),

(8.111)

(8.112)

n=0

into (8.110) gives ∞ un (x, t) = sin(x + t) − 4 cos t − π sin t − 2t sin t + 4 n=0 ∞ t + (r + s) un (r, s) drds. 0

Ω

n=0

(8.113)

8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables

281

As stated before we decompose f (x, t) into two parts as follows: f0 (x, t) = sin(x + t) − 4 cos t − π sin t, (8.114) f1 (x, t) = −2t sin t + 4. The modiﬁed decomposition technique admits the use of the recursive relation u0 (x, t) = sin(x + t) − 4 cos t − π sin t, t (r + s) (u0 (r, s)) drds u1 (x, t) = −2t sin t + 4 + (8.115) Ω 0 t uk+1 (x, t) = (r + s)uk (r, s)drds, k 1. 0

This gives

Ω

u0 (x, t) = sin(x + t) − 4 cos t − π sin t, u1 (x, t) = 4 cos t + π sin t + · · · .

(8.116)

By canceling the two noise terms from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the equation (8.110), the exact solution u(x, t) = sin(x + t), (8.117) is readily obtained. Example 8.20 Use the modiﬁed decomposition method to solve the mixed Volterra-Fredholm integral equation t F (x, t, r, s)u(r, s)drds, (x, t) ∈ Ω × [0, T ], (8.118) u(x, t) = f (x, t) + 0

where

Ω

3 + 4x 1 + 2x 3 1 4 t + t2 − t + t , 12 6 6 F (x, t, r, s) = x + t + r − 2s,

f (x, t) = x2 −

(8.119)

with Ω = [0, 1]. Proceeding as before we set ∞ 3 + 4x 1 + 2x 3 1 4 t + t2 − t + t un (x, t) = x2 − 12 6 6 n=0

t + 0

Ω

(x + t + r − 2s)

∞

un (r, s) drds. (8.120)

n=0

Decomposing f (x, t) into two parts as follows: 3 + 4x 1 + 2x 3 1 4 t + t2 , f1 (x, t) = − t + t , f0 (x, t) = x2 − 12 6 6 and using recursive relation

(8.121)

282

8 Volterra-Fredholm Integral Equations

3 + 4x t + t2 , u0 (x, t) = x2 − 12 t u1 (x, t) = v + (x + t + r − 2s)u0 (r, s)drds t uk+1 (x, t) =

0

Ω

0

Ω

(x + t + r − 2s)uk (r, s)drds,

(8.122)

k 1.

This gives 3 + 4x t + t2 , 12 (8.123) 3 + 4x u1 (x, t) = t+ ··· . 12 By canceling the noise term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the equation (8.118), the exact solution u0 (x, t) = x2 −

u(x, t) = x2 + t2 ,

(8.124)

is readily obtained.

Exercises 8.4.1 Use the modiﬁed decomposition method to solve the mixed Volterra-Fredholm integral equations in two variables t F (x, t, r, s)u(r, s)drds, (x, t) ∈ Ω × [0, T ], (8.125) u(x, t) = f (x, t) + 0

Ω

where f (x, t), F (x, t, r, s), and Ω are given by: 1. f = xt −

1 3 1 3 xt + t , F = (x − r)(t − s), Ω = [0, 1] 12 18

2. f = x + t − 3t2 − 4t3 , F = 12s, Ω = [0, 1] 2 π π π2 − t2 + 2t sin t − , F = cos r sin s, Ω = 0, 3. f = x2 − t2 + cos t 4 4 2 2 π π 4. f = x2 − t2 − sin t − t2 + 2t cos t, F = cos r cos s, Ω = 0, 4 2 1 1 t5 x, F = (x − r)(t − s), Ω = [−1, 1] 5. f = 1 + x3 + t2 + t3 − t2 + 5 10 4 3 t + t2 , F = r − s, Ω = [−1, 1] 9 π 7. f = e−t cos x + te−t cos x, F = − cos(x − r)es−t , Ω = [0, π] 2

6. f = x2 t − xt2 +

8. f = cos(x − t) + 4 sin t + cos t(π − 2t) − π, F = r − s, Ω = [0, π] 9. f = xet −

5 t 1 5 e + tet + , F = r − s, Ω = [0, 1] 6 2 6

10. f = cos(x + t) + 2π sin t − 2πt cos t, F = rs, Ω = [−π, π] 11. f = cos x cos t + 2 cos t + 2t sin t − 2, F = rs, Ω = [0, π]

References 12. f = e−t sin x +

283 π −t te sin x, F = − cos(x − r)es−t , Ω = [0, π] 2

References 1. K. Maleknejad and M. Hadizadeh, A New computational method for VolterraFredholm integral equations, Comput. Math. Appl., 37 (1999) 1–8. 2. A.M. Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. Comput., 127 (2002) 405–414. 3. H.T. Davis, Introduction to Nonlinear Diﬀerential and Integral Equations, Dover, New York, (1962). 4. R. Kress, Linear Integral Equations, Springer, Berlin, (1999). 5. G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994). 6. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, San Diego, (1986). 7. A.M.Wazwaz, A reliable modiﬁcation of Adomian decomposition method, Appl. Math. Comput., 92 (1998) 1–7. 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).

Chapter 9

Volterra-Fredholm Integro-Diﬀerential Equations

9.1 Introduction The Volterra-Fredholm integro-diﬀerential equations [1–4] appear in two types, namely: x b (k) K1 (x, t)u(t)dt + λ2 K2 (x, t)u(t)dt, (9.1) u (x) = f (x) + λ1 a

and the mixed form u

(k)

a

x b

(x) = f (x) + λ

K(r, t)u(t)dtdr, 0

(9.2)

a

k

u(x) where u(k) (x) = d dx k . The ﬁrst type contains disjoint integrals and the second type contains mixed integrals such that the Fredholm integral is the interior one, and Volterra is the exterior integral.

9.2 The Volterra-Fredholm Integro-Diﬀerential Equation In this section we will ﬁrst study the Volterra-Fredholm integro-diﬀerential equation (9.1). The Adomian decomposition method can be used after integrating both sides of any equation k times and using the initial conditions. This type of equations will be handled by using the Taylor series solution method and the variational iteration method only.

9.2.1 The Series Solution Method The series solution method [2,5] was examined before in this chapter and in Chapters 3, 4, and 5. The generic form of Taylor series at x = 0 can be A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

286

9 Volterra-Fredholm Integro-Diﬀerential Equations

written as u(x) =

∞

a n xn .

(9.3)

n=0

We will assume that the solution u(x) of the Volterra-Fredholm integrodiﬀerential equation x b K1 (x, t)u(t)dt + K2 (x, t)u(t)dt, (9.4) u(k) (x) = f (x) + 0

a

is analytic, and therefore possesses a Taylor series of the form given in (9.3), where the coeﬃcients an will be determined algebraically. In this method, we usually substitute the Taylor series (9.3) into both sides of (9.4) to obtain ∞ ∞ (k) x n n an x = T (f (x)) + K1 (x, t) an t dt n=0

0

b

K2 (x, t)

+ a

∞

n=0

an t

n

(9.5)

dt,

k=0

or for simplicity we use (a0 + a1 x + a2 x2 + · · · )(k) = T (f (x)) x

K1 (x, t) a0 + a1 t + a2 t2 + · · · dt + 0

+

b

(9.6)

K2 (x, t) a0 + a1 t + a2 t2 + · · · dt,

a

where T (f (x)) is the Taylor series for f (x). The integro-diﬀerential equation (9.4) will be converted to a traditional integral, where terms of the form tn , n 0 will be integrated. We ﬁrst integrate the right side of the integrals in (9.5) or (9.6), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine the coeﬃcients aj , j 0. Solving the resulting equations will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (9.3). The exact solution may be obtained if such an exact solution exists. It is to be noted that the series method works eﬀectively if the solution u(x) is a polynomial. However, if u(x) is any elementary function, more terms of the series are needed to obtain an approximation of high degree of accuracy. This analysis will be tested by discussing the following examples. Example 9.1 Solve the Volterra-Fredholm integro-diﬀerential equation by using the series solution method

9.2 The Volterra-Fredholm Integro-Diﬀerential Equation

u (x) = 11 + 17x − 2x3 − 3x4 +

x

1

tu(t)dt + 0

0

287

(x − t)u(t)dt, u(0) = 0. (9.7)

Substituting u(x) by the series u(x) =

∞

an xn ,

(9.8)

n=0

into both sides of the equation (9.7), and using the initial condition a0 = 0, we ﬁnd ∞ x ∞ n 3 4 n nan x = 11 + 17x − 2x − 3x + an t t dt (9.9) 0

n=1

1

+ 0

(x − t)

∞

n=0

an t

n

dt.

n=0

Evaluating the integrals at the right side, using few terms from both sides, and collecting the coeﬃcients of like powers of x, and equating the coeﬃcients of like powers of x in both sides of the resulting equation, we obtain a0 = 0, a1 = 6, a2 = 12, aj = 0, j 3. (9.10) The exact solution is therefore given by u(x) = 6x + 12x2 .

(9.11)

Example 9.2 Solve the Volterra-Fredholm integro-diﬀerential equation by using the series solution method x 1 2 3 5 u(t)dt + (1 + xt)u(t)dt, u(0) = 1. u (x) = 2 − 5x − 3x − 20x − x + 0

−1

(9.12)

Substituting u(x) by the series u(x) =

∞

an xn ,

(9.13)

n=0

into both sides of Eq. (9.12), and using the initial condition a0 = 1, we ﬁnd ∞ x ∞ n n 2 3 5 nan x = 2 − 5x − 3x − 20x − x + an t dt n=1

0

1

+

(1 + xt) −1

∞

n=0

(9.14)

n

an t dt.

n=0

Evaluating the integrals at the right side, using few terms from both sides, and collecting the coeﬃcients of like powers of x, and equating the coeﬃcients of like powers of x in both sides of the resulting equation, we obtain a0 = 1, a1 = 6, a2 = a3 = 0, a4 = 5, aj = 0, j 5. (9.15) The exact solution is therefore given by

288

9 Volterra-Fredholm Integro-Diﬀerential Equations

u(x) = 1 + 6x + 5x4 .

(9.16)

Example 9.3 Solve the Volterra-Fredholm integro-diﬀerential equation by using the series solution method x 1 u (x) = 2ex − 2 + u(t)dt + u(t)dt, u(0) = 0. (9.17) 0

0

Substituting u(x) by the Taylor polynomial 10 u(x) = an xn ,

(9.18)

n=0

and proceeding as before, we obtain a0 = 0,

a1 = 1,

a2 = 1,

a3 =

1 , 2!

a4 =

1 , 3!

a5 =

1 . 4!

(9.19)

The exact solution is given by u(x) = xex .

(9.20)

Example 9.4 Solve the Volterra-Fredholm integro-diﬀerential equation by using the series solution method x π2 u (x) = 2 sin x − x − 3 (x − t)u(t)dt + u(t)dt, (9.21) 0 0 u(0) = u (0) = 1,

u (0) = −1.

Using the series assumption for u(x) and proceeding as before, we obtain 1 a0 = 1, a1 = 1, a2 = − , 2! (9.22) 1 1 1 a 3 = − , a 4 = , a5 = , 3! 4! 5! and so on. the exact solution is given by u(x) = sin x + cos x.

(9.23)

Exercises 9.2.1 Solve the Volterra-Fredholm integro-diﬀerential equations by using the series solution method x 1 1. u (x) = 6 + 4x − x3 + (x − t)u(t)dt + (1 − xt)u(t)dt, u(0) = 0 0

−1

1 11 2 (t − 2)u(t)dt + (1 − xt)u(t)dt, u(0) = 1 x − 2x3 + 2 0 −1 x 1 3. u (x) = −4 + 6x − x2 − x4 + xu(t)dt + (1 − xt)u(t)dt, u(0) = 1

2. u (x) = 4 + 6x +

0

x

−1

9.2 The Volterra-Fredholm Integro-Diﬀerential Equation 4. u (x) = −4 + 6x + u(0) = 1 5. u (x) = −8 −

1 2 3 x + x3 + x4 + 2 4

u(0) = 1, u (0) = −3 6. u (x) = −500 +

1 4 3 6 x + x + 6 20

250 3 x + 3

x

0

x

0

u(t)dt +

0

tu(t)dt +

0

x

u(t)dt +

0

1

0

π 2

−1

1 −1

(1 − xt)u(t)dt,

(x − t)u(t)dt,

(xt2 − x2 t)u(t)dt,

tu(t)dt, u(0) = 1, u (0) = 1

9. u (x) = −6 − 2x − 3x2 − 4x3 +

x

0

u(t)dt +

u(0) = 2, u (0) = 6, u (0) = −24 x tu(t)dt + 10. u (x) = x − 2x3 + 3x4 + 0

0

0

x

0

π

u(t)dt +

tu(t)dt,

0

xu(t)dt,

u(t)dt, u(0) = 0, u (0) = 1, u (0) = 0 π 2

0

1

1

0

u(0) = 0, u (0) = 6, u (0) = −24 x 11. u (x) = 3 − 4 cos x + u(t)dt + 12. u (x) = −1 − 4 sin x +

1

u(t)dt, u(0) = −1, u (0) = 1

0

0

1

x

u(0) = 69, u (0) = −168 x u(t)dt + 7. u (x) = 2 cos x − 1 + 8. u (x) =

(xt2 − x2 t)u(t)dt +

289

u(t)dt, u(0) = 0, u (0) = 0, u (0) = 2

9.2.2 The Variational Iteration Method The variational iteration method was used before to handle Volterra and Fredholm integral equations. It was discussed before, that the method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists. The variational iteration method will also be used in this section to study the Volterra-Fredholm integro-diﬀerential equations. The standard ith order Volterra-Fredholm integro-diﬀerential equation is of the form x b u(i) (x) = f (x) + K1 (x, t)u(t)dt + K2 (x, t)u(t)dt, (9.24) 0

(i)

di u dxi ,

a

(i−1)

and u(0), u (0), . . . , u (0) are the initial conditions. where u (x) = The right side contains two disjoint integrals. Concerning the kernel K2 (x, t), we will discuss K2 (x, t) being separable and given by K2 (x, t) = g(x)h(t),

(9.25)

290

9 Volterra-Fredholm Integro-Diﬀerential Equations

or may be diﬀerence kernel and given by (9.26) K2 (x, t) = K2 (x − t) = g(x) − h(t). Consequently, the second integral at the right side of (9.24) becomes b K2 (x, t)u(t)dt = αg(x), (9.27) a

and

b

K2 (x, t)u(t)dt = βg(x) − α,

(9.28)

a

by using (9.25) and (9.26) respectively, where b α= h(t)u(t)dt, β = a

b

u(t)dt.

(9.29)

a

The correction functional for the Volterra-Fredholm integro-diﬀerential equation (9.24) is un+1 (x) = un(x) t x (9.30) (i) λ(ξ) un (ξ) − f (ξ) − K1 (ξ, r)˜ un (r)dr − αg(ξ) dξ, + 0

or

0

un+1 (x) = un (x) x t (9.31) (i) + λ(ξ) un (ξ) − f (ξ) − K1 (ξ, r)˜ un (r)dr − βg(ξ) + α dξ, 0

0

by using (9.27) and (9.28) respectively. The variational iteration method is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ(ξ) that can be identiﬁed optimally via integration by parts and by using a restricted variation. Having λ(ξ) determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), n 0 of the solution u(x). The zeroth approximation u0 can be any selective function. However, the given initial values u(0), u (0), . . . are preferably used for selecting the zeroth approximation u0 as will be seen later. Consequently, the solution is given by u(x) = lim un(x). (9.32) n→∞

It is worth noting to summarize the Lagrange multipliers λ(ξ) for a variety of ODEs as formally derived in Chapter 3: u + f (u(ξ), u (ξ)) = 0, λ = −1, u + f (u(ξ), u (ξ), u (ξ)) = 0, λ = ξ − x, 1 (ξ − x)2 , 2! 1 u(iv) + f (u(ξ), u (ξ), u (ξ), u (ξ), u(iv) (ξ)) = 0, λ = (ξ − x)3 , 3! u + f (u(ξ), u (ξ), u (ξ), u (ξ)) = 0, λ = −

9.2 The Volterra-Fredholm Integro-Diﬀerential Equation

291

1 (ξ − x)(n−1) . (n − 1)! (9.33) The VIM will be illustrated by studying the following examples. u(n) + f (u(ξ), u (ξ), u (ξ), . . . , u(n) (ξ)) = 0, λ = (−1)n

Example 9.5 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the variational iteration method x 1 u (x) = 1 + (x − t)u(t)dt + xtu(t), u(0) = 1. (9.34) 0

0

The correction functional for this equation is given by x t (t − r)un (r) dr − αt dt, un (t) − 1 − un+1 (x) = un (x) − 0

(9.35)

0

where we used λ = −1 for ﬁrst-order integro-diﬀerential equation, and 1 α= tu(t)dt. (9.36) 0

We can use the initial condition to select u0 (x) = u(0) = 1. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1, x t (t − r)u0 (r)dr − αt dt u1 (x) = u0 (x) − u0 (t) − 1 − 0

0

0

0

0

0

0

0

1 1 = 1 + x + αx2 + x3 , 2 3! x t u2 (x) = u1 (x) − (t − r)u1 (r)dr − αt dt u1 (t) − 1 − 1 1 1 1 1 = 1 + x + αx2 + x3 + x4 + αx5 + x6 , 2 3! 4! 5! 6! x t u3 (x) = u2 (x) − (t − r)u2 (r)dr − αt dt u2 (t) − 1 −

(9.37)

1 1 1 1 1 = 1 + x + αx2 + x3 + x4 + αx5 + x6 2 3! 4! 5! 6! 1 7 1 8 1 9 + x + αx + x , 7! 8! 9! x t u4 (x) = u3 (x) − (t − r)u3 (r)dr − αt dt u3 (t) − 1 − 1 1 1 1 1 = 1 + x + αx2 + x3 + x4 + αx5 + x6 2 3! 4! 5! 6! 1 1 1 1 10 1 1 12 x + αx11 + x , + x7 + αx8 + x9 + 7! 8! 9! 10! 11! 12! and so on. To determine α, we substitute u4 (x) into (9.36) to ﬁnd that α = 1.

(9.38)

292

9 Volterra-Fredholm Integro-Diﬀerential Equations

Substituting α = 1 into u4 (x), and using u(x) = lim un(x), n→∞

we ﬁnd that the exact solution is u(x) = ex ,

(9.39) (9.40)

x

obtained upon using the Taylor series for e . Example 9.6 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the variational iteration method x 1 (x − t)u(t)dt + (x − t)u(t), u(0) = 2, (9.41) u (x) = 9 − 5x − x2 − x3 + 0

0

that can be written as

2

3

u (x) = 9 − 5x − x − x + where

α=

0

x

(x − t)u(t)dt + αx − β, u(0) = 2,

1

u(t)dt,

1

β=

0

(9.42)

tu(t)dt.

(9.43)

0

The correction functional for the Volterra-Fredholm integro-diﬀerential equation is given by un+1 (x) = un (x) t x (9.44) 2 3 (t − r)un (r)dr − αt + β dt, un (t) − 9 + 5t + t + t − − 0

0

where we used λ = −1 for ﬁrst-order integro-diﬀerential equations. We can use the initial condition to select u0 (x) = u(0) = 2. Using this selection we obtain u0 (x) = 2, u1 (x) = u0 (x) t x 2 3 (t − r)u0 (r)dr − αt + β dt u0 (t) − 9 + 5t + t + t − − 0

0

5−α 2 1 4 = 2 + (9 − β)x − x − x , 2 4 u2 (x) = u1 (x) x t 2 3 − u1 (t) − 9 + 5t + t + t − (t − r)u1 (r)dr − αt + β dt 0

= 2 + (9 − β)x −

0

1 7 5−α 2 3−β 4 5−α 5 x + x − x − x , 2! 4! 5! 840

u3 (x) = u2 (x) t x 2 3 u2 (t) − 9 + 5t + t + t − (t − r)u2 (r)dr − αt + β dt, − 0

5−α 2 3−β 4 x + x = 2 + (9 − β)x − 2! 4!

0

9.2 The Volterra-Fredholm Integro-Diﬀerential Equation

293

5−α 5 3−β 7 5−α 8 (9.45) x + x − x + ··· , 5! 7! 8! and so on. To determine α and β, we substitute u3 (x) into (9.43), and solve the resulting equations to ﬁnd that α = 5, β = 3. (9.46) This in turn gives the exact solution −

u(x) = 2 + 6x, obtained upon substituting α = 5, and β = 3 into un (x).

(9.47)

Example 9.7 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the variational iteration method x 1 2 3 u(t)dt+ (1−2xt)u(t), u(0) = 2, u (0) = 6, u (x) = −8+6x−3x +x + 0

that can be written as

2

3

u (x) = −8+6x−3x +x + where

−1

x

u(t)dt+α−βx, u(0) = 2, u (0) = 6, (9.49)

0

1

u(t)dt,

α= 0

(9.48)

β=

1

2tu(t)dt.

(9.50)

0

The correction functional for the Volterra-Fredholm integro-diﬀerential equation is given by un+1 (x) = un(x) t x 2 3 (t − x) un (t) + 8 − 6t + 3t − t − un (r)dr − α + βt dt, + 0

0

(9.51) where we used λ = (t − x) for second-order integro-diﬀerential equations. We can select u0 (x) = u(0) + u (0)x = 2 + 6x to determine the successive approximations u0 (x) = 2 + 6x,

1 4 1 1 α − 4 x2 + − β x3 + x5 , 2 3 6 20 1 1 4 1 u2 (x) = 2 + 6x + α − 4 x2 + − β x3 + α− 2 3 6 120 1 1 8 1 + − β x6 + x , 90 720 6720 1 4 1 1 u3 (x) = 2 + 6x + α − 4 x2 + − β x3 + α− 2 3 6 120 1 1 1 1 − β x6 + α− x8 + 90 720 40320 20160 u1 (x) = 2 + 6x +

1 60

x5 (9.52)

1 60

x5

294

9 Volterra-Fredholm Integro-Diﬀerential Equations

+

1 1 − β x9 + · · · , 45360 362880

and so on. To determine α and β, we substitute u3 (x) into (9.50), and solve the resulting equations to ﬁnd that α = 2, β = 8. (9.53) This in turn gives the exact solution u(x) = 2 + 6x − 3x2 .

(9.54)

Example 9.8 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the variational iteration x method π 1 2 u (x) = − x + u(t)dt + xu(t), u(0) = u (0) = −u (0) = 1, (9.55) 2 0 −π that can be written as x 1 u (x) = − x2 + u(t)dt + αx, u(0) = u (0) = −u (0) = 1, (9.56) 2 0 where π α= u(t)dt. (9.57) −π

The correction functional for the Volterra-Fredholm integro-diﬀerential equation is given by t 1 x 1 2 2 (t − x) un (t) + t − un (r) dr − αt dt, un+1 (x) = un (x) − 2 0 2 0 (9.58) where we used 1 (9.59) λ = − (t − x)2 , 2 for third-order integro-diﬀerential equations. We can use the initial conditions to select 1 u0 (x) = u(0) + u (0)x + u (0)x2 , 2 1 (9.60) = 1 + x − x2 . 2 Using this selection into the correction functional gives the following successive approximations 1 u0 (x) = 1 + x − x2 , 2! 1 1 1 u1 (x) = 1 + x − x2 + (1 + α)x4 − x6 , 2 4! 6! 1 2 1 1 1 1 10 4 u2 (x) = 1 + x − x + (1 + α)x − x6 + (1 + α)x8 − x , 2 4! 6! 8! 10!

9.2 The Volterra-Fredholm Integro-Diﬀerential Equation

295

1 1 1 u3 (x) = 1 + x − x2 + (1 + α)x4 − x6 2 4! 6! 1 1 10 1 1 14 (1 + α)x8 − x + (1 + α)x12 − x , (9.61) 8! 10! 12! 14! where other approximations are obtained up to u8 (x), but not listed. To determine α, we substitute u8 (x) into (9.57), and solve the resulting equations to ﬁnd that α = 0. (9.62) +

This in turn gives the series solution 1 2 1 4 1 6 1 8 1 10 x + ··· , u(x) = x + 1 − x + x − x + x + 2 4! 6! 8! 10! that converges to the exact solution u(x) = x + cos x.

(9.63)

(9.64)

Exercises 9.2.2 Solve the following Volterra-Fredholm integro-diﬀerential equations by using the variational iteration method x 1 1. u (x) = 6 + 4x − x3 + (x − t)u(t)dt + (1 − xt)u(t)dt, u(0) = 0 0

1 2. u (x) = 6 − 2x − x2 − x3 + 2 3. u (x) = −4 + 6x − x2 − x4 + 4. u (x) = 5x −

1 3 3 x + x5 + 2 4

5. u (x) = −8 −

−1

x 0

0

x

0 x

(x − t)u(t)dt + xu(t)dt +

1

−1

−1

x

0

tu(t)dt +

xu(t)dt, u(0) = 1

(1 − xt)u(t)dt, u(0) = 1

(1 − xt)u(t)dt +

1 2 3 x + x3 + x4 + 2 4

1

1

−1 1

−1

xtu(t)dt, u(0) = 1

(x − t)u(t)dt,

u(0) = 1, u (0) = −3

x 1 2 1 xu(t)dt + (x − t)u(t)dt, u(0) = 3, u (0) = 0 − 3x2 − x5 + 5 4 0 0 x π 1 7. u (x) = −x − x3 + (x − t)u(t)dt + xu(t)dt, u(0) = 0, u (0) = 2 6 0 −π x π (x − t)u(t)dt + xu(t)dt, u(0) = 1, u (0) = 1 8. u (x) = −1 − x +

6. u (x) =

0

x

−π

1

1 1 u(t)dt + tu(t)dt, u(0) = 1, u (0) = 2, u (0) = 1 − x2 + 3 2 0 0 x π 1 10. u (x) = 5 − x4 + u(t)dt + xu(t)dt, u(0) = 0, u (0) = 1, u (0) = 0 4 0 −π x π 11. u (x) = 3 − 4 cos x + u(t)dt + u(t)dt, u(0) = 0, u (0) = 1, u (0) = 0 9. u (x) = −

0

0

296

9 Volterra-Fredholm Integro-Diﬀerential Equations

12. u (x) = −1 − 4 sin x +

x

0

u(t)dt +

π 2

0

u(t)dt, u(0) = 0, u (0) = 0, u (0) = 2

9.3 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations In a parallel manner to our analysis in the previous section, we will study the mixed Volterra-Fredholm integro-diﬀerential equation of the form x b u(i) (x) = f (x) + λ K(r, t)u(t)dtdr, (9.65) 0

a i

where f (x), K(x, t) are analytic functions, and u(i) (x) = ddxui . It is interesting to note that (9.65) contains mixed Volterra and Fredholm integral equations, where the Fredholm integral is the interior integral, whereas the Volterra integral is the exterior one. Moreover, the unknown function u(x) appears inside the integral, whereas the derivative u(i) (x) appears outside the integral. This type of equations will be handled by using the direct computation method and the series solution method. Other methods exist in the literature but will not be presented in this text.

9.3.1 The Direct Computation Method The standard ith order mixed Volterra-Fredholm integro-diﬀerential equation is of the form x

u(i) (x) = f (x) +

b

K(r, t)u(t)dtdr, 0

(9.66)

a i

where f (x), K(x, t) are analytic functions, and u(i) (x) = ddxui . The initial conditions should be prescribed. The unknown function u(x) appears inside the integral, whereas the ith derivative u(i) (x) appears outside the integral. The direct computation method [6] was used before to handle Fredholm integral equation in Chapter 4. In this section, the direct computation method will be used to solve the mixed Volterra-Fredholm integro-diﬀerential equations. The method gives the solution in an exact form and not in a series form. It is important to point out that this method will be applied for the degenerate or separable kernels of the form n K(x, t) = gk (x)hk (t). (9.67) k=1

Examples of separable kernels are x − t, xt, x2 − t2 , xt2 + x2 t, etc. We will focus our study on K(x, t) being separable of the form K(r, t) = g(r)h(t),

(9.68)

9.3 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations

297

or being diﬀerence kernel of the form K(r, t) = K(r − t) = g(r) − h(t). (9.69) Substituting (9.68) or (9.69) into (9.66), the integro-diﬀerential equation becomes x u(i) (x) = f (x) +

βg(r) dr,

(9.70)

(αg(r) − β) dr,

(9.71)

0

or u(i) (x) = f (x) +

x 0

respectively, where

b

α=

u(t)dt,

b

β=

a

h(t)u(t)dt.

(9.72)

a

Integrating (9.70) or (9.71) i times for 0 to x gives the unknown solution u(x), where the constants α and β are to be determined. This can be achieved by substituting the resulting value of u(x) into (9.72). Using the obtained numerical values of α and β, the solution u(x) of the mixed Volterra-Fredholm integro-diﬀerential equation (9.70) or (9.71) is readily obtained. The direct computation method will be illustrated by studying the following mixed Volterra-Fredholm integro-diﬀerential examples. Example 9.9 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the direct computation method x 1 1 ru(t)dtdr, u(0) = 0. (9.73) u (x) = ex (1 + x) − x2 + 2 0 0 This equation can be written as x 1 u (x) = ex (1 + x) − x2 + αrdr, u(0) = 0. (9.74) 2 0 where we used 1

α=

u(t)dt.

(9.75)

0

Integrating both sides once from 0 to x, and using the initial condition we ﬁnd α−1 3 x . u(x) = xex + (9.76) 6 To determine α, we substitute u(x) from (9.76) into (9.75) to ﬁnd that α = 1. (9.77) This in turn gives the exact solution by u(x) = xex .

(9.78)

298

9 Volterra-Fredholm Integro-Diﬀerential Equations

Example 9.10 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the direct computation method x 1 7 2 u (x) = 6 + 29x − x + (r − t)u(t)dtdr, u(0) = 0. (9.79) 2 0 0 This equation can be written as x 7 2 (αr − β)dr, (9.80) u (x) = 6 + 29x − x + 2 0 where 1

1

u(t)dt,

α=

β=

0

tu(t)dt.

(9.81)

0

Integrating both sides once from 0 to x, and using the initial condition we ﬁnd 29 − β 2 α − 1 3 x + x . u(x) = 6x + (9.82) 2 6 To determine α and β, we substitute u(x) from (9.82) into (9.81) to ﬁnd that α = 1, β = 5. (9.83) Substituting α = 7 and β = 5 into u(x) gives the exact solution by u(x) = 6x + 12x2 .

(9.84)

Example 9.11 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the direct computation method x π ru(t)dt dr, u(0) = 1, u (0) = 1. (9.85) u (x) = −x2 − sin x − cos x + 0

0

This equation can be written as u (x) = −x2 − sin x − cos x +

where

x

αr dr,

(9.86)

0

π

u(t)dt.

α=

(9.87)

0

Integrating both sides from 0 to x twice, and using the initial conditions we obtain α−2 4 x . u(x) = sin x + cos x + (9.88) 24 To determine α, we substitute u(x) into (9.87) to ﬁnd that α = 2. Substituting α = 2 into u(x) gives the exact solution by u(x) = sin x + cos x.

(9.89) (9.90)

9.3 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations

299

Example 9.12 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the direct computation method x 1 10 3 u (x) = − + 2x + (rt2 − r2 t)u(t)dt dr, u(0) = 1, u (0) = 9. (9.91) 3 0 −1 Proceeding as before, we set x 10 3 (αr − βr2 ) dr, (9.92) u (x) = − + 2x + 3 0 where 1

α=

t2 u(t)dt,

1

β=

−1

tu(t)dt.

(9.93)

−1

Integrating both sides from 0 to x twice, and using the initial conditions we obtain 5 α 6−β 5 x . u(x) = 1 + 9x − x2 + x4 + (9.94) 3 24 60 To determine α and β, we substitute u(x) into (9.93) to ﬁnd that α = 0, β = 6. Substituting α = 0 and β = 6 into u(x)gives the exact solution by 5 u(x) = 1 + 9x − x2 . 3

(9.95)

(9.96)

Exercises 9.3.1 Solve the following Volterra-Fredholm integro-diﬀerential equations by using the direct computation method x 1 5 1. u (x) = 8x + x2 + (1 − rt)u(t)dtdr, u(0) = 2 4 0 0 x 1 8 2 x + rt2 u(t)dtdr, u(0) = 1 2. u (x) = 1 + 2x − 15 0 −1 x 1 43 2 x + 3. u (x) = 1 − 2x − (1 − rt)u(t)dtdr, u(0) = 1 15 0 −1 x 1 4 3 x + 4. u (x) = 1 − 3x2 − r2 tu(t)dtdr, u(0) = 1 45 0 0 x π 2 5. u (x) = 1 − 2x − sin x + u(t)dtdr, u(0) = 1

0

x

−π 2

1 4 tu(t)dtdr, u(0) = 1 6. u (x) = 1 − x + ex + 3 0 0 x 1 1 7. u (x) = x2 − ex + rtu(t)dtdr, u(0) = 0 4 0 0 x π u(t)dtdr, u(0) = −1 8. u (x) = sin x + cos x + 0

−π

300

9 Volterra-Fredholm Integro-Diﬀerential Equations

9. u (x) = 2x3 −

1 2 x + 3

x

0

1 −1

10. u (x) = −2 sin x − x cos x + 11. u (x) = ex − 3x +

x 0

12. u (x) = −4 sin 2x +

x

0

x

0

π

−π

u(t)dtdr, u(0) = 0, u (0) = 1

tu(t)dtdr, u(0) = 2, u (0) =

0

1

(rt2 − r 2 t)u(t)dtdr, u(0) = 1, u (0) = 9

π

0

11 2

(1 + r)u(t)dtdr, u(0) = 0, u (0) = 2

9.3.2 The Series Solution Method The series solution method was examined before in this chapter. A real function u(x) is called analytic if it has derivatives of all orders such that the generic form of Taylor series at x = 0 can be written as ∞ an xn . (9.97) u(x) = n=0

In this section we will present the series solution method, that stems mainly from the Taylor series for analytic functions, for solving the mixed VolterraFredholm integro-diﬀerential equations. We will assume that the solution u(x) of the mixed Volterra-Fredholm integro-diﬀerential equation x b (i) K(r, t)u(t)dtdr, (9.98) u (x) = f (x) + 0

a

is analytic, and therefore possesses a Taylor series of the form given in (9.97), where the coeﬃcients an will be determined recurrently. In this method, we usually substitute the Taylor series (9.97) into both sides of (9.98) to obtain ∞ (i) ∞ x b k k (9.99) ak x = T (f (x)) + K(r, t) ak t dt dr, 0

k=0

a

or for simplicity we use 2

(i)

(a0 + a1 x + a2 x + · · · )

k=0

x

b

K(r, t) (a0 + a1 t + · · · ) dt dr,

= T (f (x)) + 0

a

(9.100) where T (f (x)) is the Taylor series for f (x). We integrate the inner then the outer integral in (9.99) or (9.100), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine a system of equations in aj , j 0. Solving this system will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (9.97). The exact solution may be obtained if such an

9.3 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations

301

exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. Example 9.13 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the Taylor series solution method x 1 7 u (x) = 6 + 29x − x2 + ru(t)dtdr, u(0) = 0. (9.101) 2 0 0 Substituting u(x) by the series ∞ an xn , (9.102) u(x) = n=0

into both sides of (9.101) leads to ∞ x 1 ∞ 27 2 n n (r − t) an x = 6 + 29x − x + an t dt dr. (9.103) 2 0 0 n=0 n=0 Using the initial condition gives a0 = 0. Evaluating the integrals at the right side, using few terms from both sides, collecting the coeﬃcients of like powers of x, and equating the coeﬃcients of like powers of x in both sides, we obtain a0 = 0, a1 = 6, a2 = 12, aj = 0, j 3. (9.104) The exact solution is therefore given by u(x) = 6x + 12x2 .

(9.105)

Example 9.14 Solve the Volterra-Fredholm integro-diﬀerential equation by using the series solution method x 1 x u (x) = e − x + tu(t)dtdr, u(0) = 1. (9.106) 0

0

Substituting u(x) by the series u(x) =

∞

an xn ,

(9.107)

n=0

into both sides of (9.106) leads to ∞ n an x = ex − x + n=0

0

x 1

t

0

∞

an t

n

dt dr.

(9.108)

n=0

Using the given initial condition gives a0 = 1. Evaluating the integrals at the right side, using few terms from both sides, collecting the coeﬃcients of like powers of x, and equating the coeﬃcients of like powers of x in both sides, we obtain 1 (9.109) an = , n 0. n!

302

9 Volterra-Fredholm Integro-Diﬀerential Equations

The exact solution is therefore given by u(x) = ex .

(9.110)

Example 9.15 Solve the Volterra-Fredholm integro-diﬀerential equation by using the series solution method x 1 u (x) = −2x − cos x + (1 − rt)u(t)dt dr, u(0) = 1, u (0) = 0. (9.111) 0

0

Proceeding as before, we obtain (−1)n , n 0, (2n)! and zero otherwise. This gives the exact solution by a2n =

u(x) = cos x.

(9.112)

(9.113)

Example 9.16 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the series solution method x 1 20 2 (rt2 −r2 t)u(t)dt dr, u(0) = 2, u (0) = 3. (9.114) u (x) = − + x3 + 3 3 0 −1 Using the initial conditions, and proceeding as before we ﬁnd 10 a0 = 2, a1 = 3, a2 = − , aj = 0, j 4. (9.115) 3 The exact solution is given by 10 (9.116) u(x) = 2 + 3x − x2 . 3 Exercises 9.3.2 Use the series solution method to solve the following mixed Volterra-Fredholm integrodiﬀerential equations x 1 5 1. u (x) = 8x + x2 + (1 − rt)u(t)dtdr, u(0) = 2 4 0 0 x 1 127 2. u (x) = 1 − x+ (1 − rt)u(t)dtdr, u(0) = 1 8 0 0 x 1 43 2 x + (1 − rt)u(t)dtdr, u(0) = 1 3. u (x) = 1 − 2x − 15 0 −1 x 1 77 2 x + 4. u (x) = 1 − 2x − (rt2 − r2 t)u(t)dtdr, u(0) = 1 15 0 −1 x π 2 u(t)dtdr, u(0) = 1 5. u (x) = 1 − 2x − sin x + 0

−π 2

9.4 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations 6. u (x) = ex −

3 x+ 2

x

0

1 2 x − ex + 4

8. u (x) =

1 2 x − e−x + 2

10. u (x) = −

1

x

0

1 2 x + 3

tu(t)dtdr, u(0) = 2

0

7. u (x) =

9. u (x) = 2x3 −

1

rtu(t)dtdr, u(0) = 0

0

x 0

0

−1

x

0

10 2 + x3 + 3 9

303

1 −1

x 0

rtu(t)dtdr, u(0) = 1 (rt2 − r 2 t)u(t)dtdr, u(0) = 1, u (0) = 9 1

−1

(rt2 − r2 t)u(t)dtdr, u(0) = 1, u (0) = 1

x 1 77 2 x + rtu(t)dtdr, u(0) = 1, u (0) = 1 120 0 0 x 1 rtu(t)dtdr, u(0) = 1, u (0) = 0 12. u (x) = −15x + 11. u (x) = 2 + 6x −

0

0

9.4 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations in Two Variables In this section we will study the linear mixed Volterra-Fredholm integrodiﬀerential equations in two variables given by t u (x, t) = f˜(x, t) + F (x, t, r, s)u(r, s)drds, (x, t) ∈ Ω × [0, T ], (9.117) 0

Ω

where u(0, t) = u0 . The functions f˜(x), and F (x, t, r, s) are analytic functions on D = Ω×[0, T ], and Ω is a closed subset of Rn , n = 1, 2, 3. It is interesting to note that (9.117) contains mixed Volterra and Fredholm integral equations, where the Fredholm integral is the interior integral, whereas the Volterra integral is the exterior one. Moreover, the unknown function u(x, t) appears inside the integral, whereas the derivative u (x) appears outside the integral. The mixed Volterra-Fredholm integro-diﬀerential equation (9.117) arises from parabolic boundary value problems, and in various physical and biological models. In the literature, some methods, such as the projection method, time collocation method, the trapezoidal Nystrom method, Adomian method, and other analytical or numerical techniques were used to handle this equation. It was found that these techniques encountered diﬃculties in terms of computational work used, and therefore, approximate solutions were obtained for numerical purposes. In particular, it was found that a complicated term f (x, t) can cause diﬃcult integrations and proliferation of terms in Adomian recursive scheme. To overcome the tedious work of the existing strategies, the modiﬁed decomposition method, combined sometimes with the noise terms phenomenon, will form a useful basis for studying the mixed Volterra-Fredholm integro-

304

9 Volterra-Fredholm Integro-Diﬀerential Equations

diﬀerential equation (9.117). The size of the computational work can be dramatically reduced by using the modiﬁed decomposition method. The noise terms were deﬁned as the identical terms with opposite signs that may appear in the ﬁrst two components of the series solution of u(x, t). The modiﬁed decomposition method and the noise terms phenomenon were presented in details in Chapters 3, 4 and in this chapter.

9.4.1 The Modiﬁed Decomposition Method We ﬁrst integrate both sides of (9.117) from 0 to x to obtain x t F (w, t, r, s)u(r, s) dr ds dw, u(x, t) − u(0, t) = f (x, t) + 0

where

(9.118)

Ω

0

x

f˜(w, t)dw,

f (x, t) =

(9.119)

0

and (x, t) ∈ Ω × [0, T ], u(0, t) = u0 . For mixed integro-diﬀerential equations of higher orders, we can integrate the equation as many times as the order of the given equation. However, in this section we will focus our study only on the ﬁrst order equations. The modiﬁed decomposition method expresses the function f (x, t) as the sum of two partial functions, namely f1 (x, t) and f2 (x, t). In other words, we can set f (x, t) = f1 (x, t) + f2 (x, t). (9.120) To minimize the size of calculations, we identify the zeroth component u0 (x, t) by one part of f (x, t), namely f1 (x, t) or f2 (x, t). The other part of f (x, t) can be added to the component u1 (x, t) among other terms. In other words, the modiﬁed decomposition method [7–8] introduces the modiﬁed recurrence relation u0 (x, t) = f1 (x, t), x t u1 (x, t) = f2 (x, t) + F (w, t, r, s)u0 (r, s)drdsdw, (9.121) 0 0 Ω x t F (w, t, r, s)uk (r, s)drdsdw, k 1. uk+1 (x, t) = 0

0

Ω

If noise terms appear between u0 (x, t) and u1 (x, t), then by canceling these terms from u0 (x, t), the remaining non-canceled terms of u0 (x, t) may give the exact solution. This can be satisﬁed by direct substitution. In what follows, we study some illustrative examples. Example 9.17 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the modiﬁed decomposition method

9.4 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations

u (x, t) = t +

1 3 1 t − xt3 + 18 12

305

t 0

Ω

(x − r)(t − s)u(r, s)drds, u(0, t) = 0, (9.122)

with Ω = [0, 1]. Substituting the decomposition series ∞ u(x, t) = un (x, t),

(9.123)

n=0

into (9.122) gives ∞ 1 1 un (x, t) = t + t3 − xt3 18 12 n=0 ∞ t + (x − r)(t − s) un (r, s) drds. Ω

0

(9.124)

n=0

Integrating both sides from 0 to x and using the initial condition we ﬁnd ∞ 1 1 un (x, t) = xt + xt3 − x2 t3 18 24 n=0 ∞ (9.125) x t + (w − t)(r − s) un (r, s) drdsdw. 0

0

Ω

n=0

We then decompose f (x, t) into two parts as follows: 1 1 f0 (x, t) = xt + xt3 , f1 (x, t) = − x2 t3 . (9.126) 18 24 The modiﬁed decomposition technique admits the use of the recursive relation 1 u0 (x, t) = xt + xt3 , 18 x t 1 2 3 u1 (x, t) = − x t + (w − t)(r − s)u0 (r, s)drdsdw, (9.127) 24 0 0 Ω x t (w − t)(r − s)uk (r, s)drdsdw, k 1. uk+1 (x, t) = 0

0

Ω

This gives u0 (x, t) = xt +

1 3 xt , 18

1 1 1 2 5 u1 (x, t) = − xt3 − xt5 + x t . 18 1080 1440

(9.128)

1 1 The self-canceling noise terms 18 xt3 and − 18 xt3 appear between the components u0 (x, t) and u1 (x, t) respectively. By canceling this term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the equation (9.122), the exact solution u(x, t) = xt, (9.129)

306

9 Volterra-Fredholm Integro-Diﬀerential Equations

is readily obtained. Example 9.18 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the modiﬁed decomposition method t 1 1 u (x, t) = 2x − t2 + t4 + rtu(r, s)drds, u(0, t) = −t2 , (9.130) 4 6 0 Ω with Ω = [0, 1]. Substituting the decomposition series ∞ un (x, t), (9.131) u(x, t) = n=0

into the last equation gives ∞ ∞ t 1 2 1 4 un (x, t) = 2x − t + t + rt un (r, s) drds. (9.132) 4 6 0 Ω n=0 n=0 Integrating both sides from 0 to x and using the initial condition we ﬁnd ∞ 1 1 un (x, t) = x2 − t2 − xt2 + xt4 4 6 n=0 ∞ (9.133) x t + rt un (r, s) drdsdw. 0

Ω

0

n=0

We decompose f (x, t) into two parts as follows: 1 1 (9.134) f0 (x, t) = x2 − t2 − xt2 , f1 (x, t) = xt4 . 4 6 The modiﬁed decomposition technique admits the use of the recursive relation 1 u0 (x, t) = x2 − t2 − xt2 , 4 x t 1 4 u1 (x, t) = xt + rtu0 (r, s)drdsdw, (9.135) 6 x t 0 0 Ω uk+1 (x, t) = rtuk (r, s)drdsdw, k 1. 0

0

Ω

This gives 1 u0 (x, t) = x2 − t2 − xt2 , 4 1 2 1 4 u1 (x, t) = xt − xt . 4 36

(9.136)

The self-canceling noise terms − 41 xt2 and 14 xt2 appear between the components u0 (x, t) and u1 (x, t) respectively. By canceling this term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the given equation, the exact solution is given by u(x, t) = x2 − t2 .

(9.137)

9.4 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations

307

Example 9.19 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the modiﬁed decomposition method t 1 1 t 1 2 x u (x, t) = e − + e (1 + t) − t + rsu(r, s)drds, u(0, t) = 1 + et , 2 2 2 Ω 0 (9.138) with Ω = [0, 1]. Proceeding as before, we ﬁnd ∞ 1 1 1 un (x, t) = ex + et − x + xet (1 + t) − xt2 2 2 2 n=0 ∞ (9.139) x t + rs un (r, s) drdsdw. 0

0

Ω

n=0

We decompose f (x, t) into two parts as follows: 1 f0 (x, t) = ex + et − x, 2 (9.140) 1 t 1 f1 (x, t) = xe (1 + t) − xt2 . 2 2 The modiﬁed decomposition technique admits the use of the recursive relation 1 u0 (x, t) = ex + et − x, 2 x t 1 1 u1 (x, t) = xet (1 + t) − xt2 + rsu0 (r, s)drdsdw, (9.141) 2 2 0 0 Ω x t rsuk (r, s)drdsdw, k 1. uk+1 (x, t) = 0

0

Ω

This in turn gives 1 u0 (x, t) = ex + et − x, 2 1 2 1 u1 (x, t) = x − xt . 2 12

(9.142)

The self-canceling noise terms − 21 x and 12 x appear between the components u0 (x, t) and u1 (x, t) respectively. By canceling this term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the given equation, the exact solution is therefore given by u(x, t) = ex + et .

(9.143)

Example 9.20 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the modiﬁed decomposition method t 1 3 1 3 u (x, t) = t cos x− t − xt + (xt+rs)u(r, s)drds, u(0, t) = 0, (9.144) 3 2 Ω 0 with Ω = [0, π2 ]. Proceeding as before, we ﬁnd

308

9 Volterra-Fredholm Integro-Diﬀerential Equations ∞

1 1 un (x, t) = t sin x − xt3 − x2 t3 3 4 n=0 ∞ x t + (wt + rs) un (r, s) drdsdw. (9.145) 0

0

Ω

n=0

We decompose f (x, t) into two parts as follows: 1 f0 (x, t) = t sin x − xt3 , 3 (9.146) 1 2 3 f1 (x, t) = − x t . 4 The modiﬁed decomposition technique admits the use of the recursive relation 1 u0 (x, t) = t sin x − xt3 , 3 x t 1 u1 (x, t) = − x2 t3 + (wt + rs)u0 (r, s)drdsdw, (9.147) 4 Ω 0 0 x t uk+1 (x, t) = (wt + rs)uk (r, s)drdsdw, k 1. 0

0

Ω

This in turn gives 1 u0 (x, t) = t sin x − xt3 , 3 1 3 u1 (x, t) = xt + · · · . 3

(9.148)

The self-canceling noise terms − 31 xt3 and 13 xt3 appear between the components u0 (x, t) and u1 (x, t) respectively. By canceling this term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the given equation, the exact solution is therefore given by u(x, t) = t sin x.

(9.149)

Exercises 9.4.1 Use the modiﬁed decomposition method to solve the mixed Volterra-Fredholm integro-diﬀerential equations in two variables t F (x, t, r, s)u(r, s)drds, (x, t) ∈ Ω × [0, T ], (9.150) u (x, t) = f (x, t) + 0

Ω

where f (x, t), F (x, t, r, s), u(0, t) and Ω are given by: 1 2 1 3 t − t , F = rs, u(0, t) = t, Ω = [0, 1] 6 6 1 2 1 4 2. f = 2x − t − t , F = rs, u(0, t) = t2 , Ω = [0, 1] 8 8 2 3 1 3 3. f = 2t − t − t , F = rs, u(0, t) = 1, Ω = [0, 1] 9 4 1. f = 1 −

References

309

5 2 1 3 1 t− t − t , F = r + s, u(0, t) = t, Ω = [0, 1] 4 12 3 7 4 2 3 2 t − xt , F = x + t + r + s, u(0, t) = 0, Ω = [−1, 1] 5. f = 2xt − 18 9 2 5 1 4 2 6. f = 3x + t + xt , F = x + s, u(0, t) = −t3 , Ω = [−1, 1] 5 2 4. f = 2x −

7. f = −t sin x + t2 , F = r + s, u(0, t) = 0, Ω = [0, π] π π 8. f = cos x − t2 , F = rs, u(0, t) = t, Ω = − , 2 2 π π 9. f = − sin x, F = rs, u(0, t) = 1 + sin t, Ω = − , 2 2 1 2 1 3 t − t , F = rs, u(0, t) = 1 + t, Ω = [0, 1] 2 6 1 1 11. f = et + xt − xtet , F = xt, u(0, t) = 0, Ω = [0, 1] 2 2 1 12. f = tex − t3 , F = rs, u(0, t) = t, Ω = [0, 1] 3

10. f = ex −

References 1. K. Maleknejad and M. Hadizadeh, A New computational method for VolterraFredholm integral equations, Comput. Math. Appl., 37 (1999) 1–8. 2. K. Maleknejad, and Y. Mahmoudi, Taylor polynomial solution of high order nonlinear Volterra-Fredholm integro-diﬀerential equations, Appl. Math. Comput., 145 (2003) 641–653. 3. A.M. Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. Comput., 127 (2002) 405–414. 4. S.A. Youseﬁ, A. Lotﬁ and M. Dehghan, He’s variational iteration method for solving nonlinear mixed Volterra-Fredholm integral equations, Comput. Math. Appl., 58 (2009) 2172–2176. 5. R. Kress, Linear Integral Equations, Springer, Berlin, (1999). 6. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 7. G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994). 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).

Chapter 10

Systems of Volterra Integral Equations

10.1 Introduction Systems of integral equations, linear or nonlinear, appear in scientiﬁc applications in engineering, physics, chemistry and populations growth models [1–4]. Studies of systems of integral equations have attracted much concern in applied sciences. The general ideas and the essential features of these systems are of wide applicability. The systems of Volterra integral equations appear in two kinds. For systems of Volterra integral equations of the ﬁrst kind, the unknown functions appear only under the integral sign in the form: x % 1 (x, t)v(t) + · · · dt, K1 (x, t)u(t) + K f1 (x) = 0

f2 (x) =

x

0

% 2 (x, t)v(t) + · · · dt, K2 (x, t)u(t) + K

(10.1)

.. . However, systems of Volterra integral equations of the second kind, the unknown functions appear inside and outside the integral sign of the form: x % 1 (x, t)v(t) + · · · dt, K1 (x, t)u(t) + K u(x) = f1 (x) + 0

v(x) = f2 (x) + .. .

0

x

% 2 (x, t)v(t) + · · · dt, K2 (x, t)u(t) + K

(10.2)

% i (x, t), and the functions fi (x), i = 1, 2, . . . , n are The kernels Ki (x, t) and K given real-valued functions. A variety of analytical and numerical methods are used to handle systems of Volterra integral equations. The existing techniques encountered some diﬃculties in terms of the size of computational work, especially when the system A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

312

10 Systems of Volterra Integral Equations

involves several integral equations. To avoid the diﬃculties that usually arise from the traditional methods, we will use some of the methods presented in this text. The Adomian decomposition method, the Variational iteration method, and the Laplace transform method will form a reasonable basis for studying systems of integral equations. The emphasis in this text will be on the use of these methods rather than proving theoretical concepts of convergence and existence that can be found in other texts.

10.2 Systems of Volterra Integral Equations of the Second Kind We will ﬁrst study systems of Volterra integral equations of the second kind given by x % 1 (x, t)v(t) + · · · dt, u(x) = f1 (x) + K1 (x, t)u(t) + K 0 (10.3) x % 2 (x, t)v(t) + · · · dt. v(x) = f2 (x) + K2 (x, t)u(t) + K 0

The unknown functions u(x), v(x), . . ., that will be determined, appear in% i (x, t), and the side and outside the integral sign. The kernels Ki (x, t) and K function fi (x) are given real-valued functions. In what follows we will present the methods, new and traditional, that will be used to handle these systems.

10.2.1 The Adomian Decomposition Method The Adomian decomposition method [5–7] was presented before. The method decomposes each solution as an inﬁnite sum of components, where these components are determined recurrently. This method can be used in its standard form, or combined with the noise terms phenomenon. Moreover, the modiﬁed decomposition method will be used wherever it is appropriate. It is interesting to point out that the VIM method can also be used, but we need to transform the system of integral equations to a system of integro-diﬀerential equations that will be presented later in this chapter. Example 10.1 Use the Adomian decomposition method to solve the following system of Volterra integral equations

10.2 Systems of Volterra Integral Equations of the Second Kind

1 u(x) = x − x4 + 6

x

313

(x − t)2 u(t) + (x − t)v(t) dt,

0

(10.4) x

1 v(x) = x2 − x5 + (x − t)3 u(t) + (x − t)2 v(t) dt. 12 0 The Adomian decomposition method suggests that the linear terms u(x) and v(x) be decomposed by an inﬁnite series of components ∞ ∞ un (x), v(x) = vn (x), u(x) = (10.5) n=0

n=0

where un (x) and vn (x), n 0 are the components of u(x) and v(x) that will be elegantly determined in a recursive manner. Substituting (10.5) into (10.4) gives x ∞ ∞ ∞ 1 4 2 un (x) = x − x + un (t) + (x − t) vn (t) dt, (x − t) 6 0 n=0 n=0 n=0 x ∞ ∞ ∞ 1 5 2 3 2 (x − t) vn (x) = x − x + un (t) + (x − t) vn (t) dt. 12 0 n=0 n=0 n=0 (10.6) The zeroth components u0 (x) and v0 (x) are deﬁned by all terms that are not included under the integral sign. Following Adomian analysis, the system (10.6) is transformed into a set of recursive relations given by 1 u0 (x) = x − x4 , 6 x (10.7)

(x − t)2 uk (t) + (x − t)vk (t) dt, k 0, uk+1 (x) = 0

and 1 v0 (x) = x2 − x5 , 12 x

(x − t)3 uk (t) + (x − t)2 vk (t) dt, vk+1 (x) = 0

(10.8) k 0.

This in turn gives 1 u0 (x) = x − x4 , 6 and v0 (x) = x2 −

1 5 x , 12

u1 (x) = v1 (x) =

1 4 1 7 x − x , 6 280 1 5 11 8 x − x . 12 10080

(10.9) (10.10)

It is obvious that the noise terms ∓ 16 x4 appear between u0 (x) and u1 (x). 1 5 Moreover, the noise terms ∓ 12 x appear between v0 (x) and v1 (x). By canceling these noise terms from u0 (x) and v0 (x), the non-canceled terms of u0 (x) and v0 (x) give the exact solutions (10.11) (u(x), v(x)) = (x, x2 ).

314

10 Systems of Volterra Integral Equations

Example 10.2 Use the Adomian decomposition method to solve the following system of Volterra integral equations x (sin(x − t)u(t) + cos(x − t)v(t)) dt, u(x) = cos x − x sin x + 0 (10.12) x v(x) = sin x − x cos x + (cos(x − t)u(t) − sin(x − t)v(t)) dt. 0

We ﬁrst decompose the linear terms u(x) and v(x) by an inﬁnite series of components ∞ ∞ un (x), v(x) = vn (x), u(x) = (10.13) n=0

n=0

where un (x) and vn (x), n 0 are the components of u(x) and v(x) that will be elegantly determined in a recursive manner. Substituting (10.13) into (10.12) gives ∞ un (x) = cos x − x sin x n=0 x ∞ ∞ + un(t) + cos(x − t) vn (t) dt, sin(x − t) 0

∞ n=0

vn (x) = sin x − x cos x x

+ 0

cos(x − t)

n=0

n=0

∞

∞

un (t) − sin(x − t)

n=0

(10.14) vn (t) dt.

n=0

The zeroth components u0 (x) and v0 (x) are deﬁned by all terms that are not included under the integral sign. For this example, we will use the modiﬁed decomposition method, therefore we set the recursive relation u0 (x) = cos x, x (sin(x − t)uk (t) + cos(x − t)vk (t)) dt, k 0, uk+1 (x) = −x sin x + 0

(10.15)

and v0 (x) = sin x, vk+1 (x) = −x cos x + This in turn gives u0 (x) = cos x,

x

0

(cos(x − t)uk (t) − sin(x − t)vk (t)) dt,

k 0. (10.16)

u1 (x) = 0,

uk+1 (x) = 0,

k 1,

(10.17)

v1 (x) = 0,

vk+1 (x) = 0,

k 1.

(10.18)

and v0 (x) = sin x,

10.2 Systems of Volterra Integral Equations of the Second Kind

315

This gives the exact solutions (u(x), v(x)) = (cos x, sin x), that satisfy the system (10.12).

(10.19)

Example 10.3 Use the Adomian decomposition method to solve the following system of Volterra integral equations x

1 1 u(x) = 1 + x2 − x3 − x4 + (x − t)3 u(t) + (x − t)2 v(t) dt, 3 3 0 x

1 1 v(x) = 1 + x − x3 − x4 − x5 + (x − t)4 u(t) + (x − t)3 v(t) dt. 4 4 0 (10.20) Proceeding as before we ﬁnd ∞ 1 1 un (x) = 1 + x2 − x3 − x4 3 3 n=0 x ∞ ∞ 3 2 + un (t) + (x − t) vn (t) dt, (x − t) 0

∞

n=0

n=0

1 1 vn (x) = 1 + x − x3 − x4 − x5 4 4 n=0 x ∞ ∞ 4 3 (x − t) + un (t) + (x − t) vn (t) dt. 0

n=0

(10.21)

n=0

We next set the recursive relations 1 1 u0 (x) = 1 + x2 − x3 − x4 , 3 3 x

uk+1 (x) = (x − t)3 uk (t) + (x − t)2 vk (t) dt, 0

(10.22) k 0,

and 1 1 v0 (x) = 1 + x − x3 − x4 − x5 , 4 4 x

(x − t)4 uk (t) + (x − t)3 vk (t) dt, vk+1 (x) = 0

(10.23) k 0.

This in turn gives

and

1 1 u0 (x) = 1 + x2 − x3 − x4 , 3 3 1 3 1 4 u1 (x) = x + x + · · · , 3 3

(10.24)

1 1 v0 (x) = 1 + x − x3 − x4 − x5 , 4 4 1 1 v1 (x) = x4 + x5 + · · · . 4 4

(10.25)

316

10 Systems of Volterra Integral Equations

It is obvious that the noise terms ∓ 31 x3 and ∓ 31 x4 appear between u0 (x) and u1 (x). Moreover, the noise terms ∓ 14 x4 and ∓ 41 x5 appear between v0 (x) and v1 (x). By canceling these noise terms from u0 (x) and v0 (x), the non-canceled terms of u0 (x) and v0 (x) give the exact solutions (u(x), v(x)) = (1 + x2 , 1 + x − x3 ), (10.26) that satisfy the given system (10.20). Example 10.4 Use the Adomian decomposition method to solve the following system of Volterra integral equations x −t

x e u(t) + et v(t) dt, u(x) = e − 2x + 0 (10.27) x

t −t −x e u(t) + e v(t) dt. v(x) = e + sinh 2x + 0

Proceeding as before we ﬁnd x ∞ ∞ ∞ −t t x un (x) = e − 2x + un (t) + e vn (t) dt, e 0

n=0 ∞

vn (x) = e−x + sinh 2x +

n=0

n=0

x

et 0

n=0 ∞

un (t) + e−t

n=0

∞

(10.28)

vn (t) dt.

n=0

Proceeding as before we set u0 (x) = ex − 2x, and

v0 (x) = e−x + sinh 2x,

u1 (x) = 2x + · · · ,

(10.29)

v1 (x) = − sinh 2x + · · · .

(10.30)

By canceling the noise terms from u0 (x) and v0 (x), the non-canceled terms of u0 (x) and v0 (x) give the exact solutions (u(x), v(x)) = (ex , e−x ), that satisfy the given system (10.27).

(10.31)

Exercises 10.2.1 Use the Adomian decomposition method to solve the following systems of Volterra integral equations ⎧ x

1 5 ⎪ 2 ⎪ x (x − t)2 u(t) + (x − t)v(t) dt u(x) = x − + ⎪ ⎨ 12 0 1. x ⎪

1 6 ⎪ ⎪ (x − t)3 u(t) + (x − t)2 v(t) dt x + ⎩ v(x) = x3 − 30 0

10.2 Systems of Volterra Integral Equations of the Second Kind ⎧ x ⎪ 2 3 ⎪ − x + (xtu(t) + xtv(t)) dt u(x) = 1 + x + x ⎪ ⎨ 0 2. x ⎪ 1 1 ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎩ v(x) = 1 − x − x2 − x3 − x4 − 3 6 0 ⎧ x ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎪ u(x) = 1 − x2 + x3 + ⎨ 0 3. x ⎪ 1 5 ⎪ ⎪ x + ((x − t)u(t) − (x − t)v(t)) dt ⎩ v(x) = 1 − x3 − 10 0 ⎧ x

1 6 ⎪ ⎪ x + (x − t)2 u(t) − (x − t)2 v(t) dt ⎪ u(x) = x2 + x3 − ⎨ 30 0 4. x ⎪ 1 ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎩ v(x) = x2 − x3 − x4 + 6 0 ⎧ x ⎪ ⎪ (cos(x − t)u(t) + sin(x − t)v(t)) dt ⎪ u(x) = cos x − v + ⎨ 5.

0

⎪ ⎪ ⎪ ⎩ v(x) = sin x − xv +

x 0

(sin(x − t)u(t) + cos(x − t)v(t)) dt

⎧ x ⎪ ⎪ (cos(x − t)u(t) + sin(x − t)v(t)) dt u(x) = sin x − xu + ⎪ ⎨ 0 6. x ⎪ ⎪ ⎪ (sin(x − t)u(t) + cos(x − t)v(t)) dt ⎩ v(x) = cos x − u + 0

⎧ x ⎪ 2 ⎪ x − 2 tan x + x + (u(t) + v(t)) dt u(x) = sec ⎪ ⎨ 0 7. x ⎪ ⎪ ⎪ (u(t) − v(t)) dt ⎩ v(x) = tan2 x − x + 0

⎧ x 1 2 ⎪ 2 ⎪ x − + ((x − t)u(t) + (x − t)v(t)) dt u(x) = sin x ⎪ ⎨ 2 0 8. x ⎪

1 1 ⎪ ⎪ (x − t)2 u(t) − (x − t)2 v(t) dt ⎩ v(x) = cos2 x + x − sin 2x + 2 2 0 ⎧ x −t

⎪ −x ⎪ u(x) = e e u(t) + et v(t) dt − sinh 2x + ⎪ ⎨ 9.

⎪ ⎪ ⎪ ⎩ v(x) = ex − 2x +

0

x 0

t

e u(t) + e−t v(t) dt

⎧ ⎪ x ⎪ + u(x) = 1 − 2x + e ⎪ ⎨

x

(u(t) + v(t)) dt

0

10.

x ⎪ 1 ⎪ ⎪ (u(t) − tv(t)) dt ⎩ v(x) = 1 − x + x2 − (x + 1)ex + 2 0 ⎧ x ⎪ ⎪ u(x) = 1 − 2x + sin x + (u(t) + v(t)) dt ⎪ ⎨

11.

⎪ ⎪ ⎪ ⎩ v(x) = 1 − x2 − sin x +

0

x

0

(tu(t) + tv(t)) dt

317

318

10 Systems of Volterra Integral Equations ⎧ ⎪ 2 ⎪ + cos x + u(x) = x − x ⎪ ⎨

12.

x

0

⎪ 2 ⎪ ⎪ ⎩ v(x) = x − x3 − cos x + 3

(u(t) + v(t)) dt x 0

(tu(t) + tv(t)) dt

10.2.2 The Laplace Transform Method The Laplace transform method is a powerful technique that can be used for solving initial value problems and integral equations as well. The Laplace transform method was presented in Chapter 1, and has been used in other chapters in this text. Before we start applying this method, we summarize some of the concepts presented in Section 1.5. In the convolution theorem for the Laplace transform, it was stated that if the kernel K(x, t) of the integral equation x

u(x) = f (x) + λ

K(x, t)u(t)dt,

(10.32)

0

depends on the diﬀerence x − t, then it is called a diﬀerence kernel. Examples of the diﬀerence kernel are ex−t , cos(x − t), and x − t. The integral equation can thus be expressed as x K(x − t)u(t)dt. (10.33) u(x) = f (x) + λ 0

Consider two functions f1 (x) and f2 (x) that possess the conditions needed for the existence of Laplace transform for each. Let the Laplace transforms for the functions f1 (x) and f2 (x) be given by L{f1 (x)} = F1 (s), L{f2 (x)} = F2 (s). (10.34) The Laplace convolution product of these two functions is deﬁned by x (f1 ∗ f2 )(x) = f1 (x − t)f2 (t)dt, (10.35) 0

or (f2 ∗ f1 )(x) =

0

x

f2 (x − t)f1 (t)dt.

(10.36)

Recall that (f1 ∗ f2 )(x) = (f2 ∗ f1 )(x).

(10.37)

We can easily show that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x f1 (x − t)f2 (t)dt} = F1 (s)F2 (s). (10.38) L{(f1 ∗ f2 )(x)} = L{ 0

Based on this summary, we will examine speciﬁc Volterra integral equations where the kernel is a diﬀerence kernel. Recall that we will apply the Laplace transform method and the inverse of the Laplace transform using Table 1.1

10.2 Systems of Volterra Integral Equations of the Second Kind

319

in Section 1.5. The Laplace transform method for solving systems of Volterra integral equations will be illustrated by studying the following examples. Example 10.5 Solve the system of Volterra integral equations by using the Laplace transform method x

u(x) = 1 − x2 + x3 + 3

v(x) = 1 − x −

1 5 x 10

0

((x − t)u(t) + (x − t)v(t)) dt,

+ 0

(10.39)

x

((x − t)u(t) − (x − t)v(t)) dt.

Notice that the kernels K1 (x − t) = K2 (x − t) = x − t. Taking Laplace transform of both sides of each equation in (10.39) gives U (s) = L{u(x)} = L{1 − x2 + x3 } + L{(x − t) ∗ u(x) + (x − t) ∗ v(x)}, 1 V (s) = L{v(x)} = L 1 − x3 − x5 + L{(x − t) ∗ u(x) − (x − t) ∗ v(x)}. 10 (10.40) This in turn gives 2 1 6 1 1 U (s) = − 3 + 4 + 2 U (s) + 2 V (s), s s s s s (10.41) 6 12 1 1 1 V (s) = − 4 − 6 + 2 U (s) − 2 V (s), s s s s s or equivalently 1 6 1 1 2 1 − 2 U (s) − 2 V (s) = − 3 + 4 , s s s s s (10.42) 1 1 6 1 12 1 + 2 V (s) − 2 U (s) = − 4 − 6 . s s s s s Solving this system of equations for U (s) and V (s) gives 3! 1 U (s) = + 4 , s s (10.43) 3! 1 V (s) = − 4 . s s By taking the inverse Laplace transform of both sides of each equation in (10.43), the exact solutions are given by (10.44) (u(x), v(x)) = (1 + x3 , 1 − x3 ). Example 10.6 Solve the system of Volterra integral equations by using the Laplace transform method x

u(x) = cos x − sin x + v(x) = sin x − x sin x +

0 0

(cos(x − t)u(t) + sin(x − t)v(t)) dt, (10.45)

x

(sin(x − t)u(t) + cos(x − t)v(t)) dt.

320

10 Systems of Volterra Integral Equations

Taking Laplace transform of both sides of each equation in (10.45) gives U (s) = L{u(x)} = L{cos x − sin x} + L{cos(x − t) ∗ u(x) + sin(x − t) ∗ v(x)}, (10.46) V (s) = L{v(x)} = L{sin x − x sin x} + L{sin(x − t) ∗ u(x) + cos(x − t) ∗ v(x)}. This in turn gives s 1 s 1 U (s) = − + U (s) + V (s), 2 2 2 1+s 1+s 1+s 1 + s2 (10.47) 2s 1 s 1 − + U (s) + V (s), V (s) = 1 + s2 (1 + s2 )2 1 + s2 1 + s2 or equivalently 1 s s 1 U (s) − 1− V (s) = − , 1 + s2 1 + s2 1 + s2 1 + s2 (10.48) 1 2s s 1 U (s) = − . 1− V (s) − 1 + s2 1 + s2 1 + s2 (1 + s2 )2 Solving this system of equations for U (s) and V (s) gives s 1 U (s) = , V (s) = . (10.49) 1 + s2 1 + s2 By taking the inverse Laplace transform of both sides of each equation in (10.49), the exact solutions are given by (u(x), v(x)) = (cos x, sin x). (10.50) Example 10.7 Solve the system of Volterra integral equations by using the Laplace transform method u(x) = 2 − e−x +

x

((x − t)u(t) + (x − t)v(t)) dt, x v(x) = 2x − ex + 2e−x + ((x − t)u(t) − (x − t)v(t)) dt. 0

(10.51)

0

Taking Laplace transform of both sides of each equation in (10.51) gives U (s) = L{2 − e−x } + L{(x − t) ∗ u(x) + (x − t) ∗ v(x)}, (10.52) V (s) = L{2x − ex + 2e−x } + L{(x − t) ∗ u(x) − (x − t) ∗ v(x)}. This in turn gives 1 1 1 2 + 2 U (s) + 2 V (s), U (s) = − s s+1 s s (10.53) 2 2 1 1 1 V (s) = 2 − + + U (s) − 2 V (s), s s − 1 s + 1 s2 s or equivalently

10.2 Systems of Volterra Integral Equations of the Second Kind

321

1 2 1 1 , 1 − 2 U (s) − 2 V (s) = − s s s s+1 (10.54) 1 1 2 2 1 1 + 2 V (s) − 2 U (s) = 2 − + . s s s s−1 s+1 Solving this system of equations for U (s) and V (s) gives 1 1 U (s) = , V (s) = . (10.55) s−1 s+1 Taking the inverse Laplace transform of (10.55) gives the exact solutions by (u(x), v(x)) = (ex , e−x ).

(10.56)

Example 10.8 Solve the system of Volterra integral equations by using the Laplace transform method x 1 4 1 5 u(x) = x − x − x + ((x − t)v(t) + (x − t)w(t)) dt, 12 20 0x 1 1 (10.57) v(x) = x2 − x3 − x5 + ((x − t)u(t) + (x − t)w(t)) dt, 6 20 x 0 1 5 ((x − t)u(t) + (x − t)v(t)) dt. w(x) = x3 − x4 + 6 12 0 Taking Laplace transform of both sides of each equation in (10.57) gives 1 1 U (s) = L x − x4 − x5 + L {(x − t) ∗ v(x) + (x − t) ∗ w(x)} , 12 20 1 1 5 2 3 V (s) = L x − x − x + L {(x − t) ∗ u(x) + (x − t) ∗ w(x)} , 6 20 5 3 1 x − x4 + L {(x − t) ∗ u(x) + (x − t) ∗ v(x)} W (s) = L 6 12 (10.58) Proceeding as before we ﬁnd 1 1 1 2 6 U (s) − 2 V (s) − 2 W (s) = 2 − 5 − 6 , s s s s s 1 2 1 6 1 (10.59) V (s) − 2 U (s) − 2 W (s) = 3 − 4 − 6 , s s s s s 1 1 5 10 W (s) − 2 U (s) − 2 V (s) = 4 − 5 . s s s s Solving this system of equations for U (s), V (s) and W (s) gives 2! 3! 1 (10.60) U (s) = 2 , V (s) = 3 , W (s) = 4 . s s s By taking the inverse Laplace transform of both sides of each equation in (10.60), the exact solutions are given by (10.61) (u(x), v(x), w(x)) = (x, x2 , x3 ).

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10 Systems of Volterra Integral Equations

Exercises 10.2.2 Use the Laplace transform method to solve the following systems of Volterra integral equations ⎧ x

1 5 ⎪ 2 − ⎪ x (x − t)2 u(t) + (x − t)v(t) dt u(x) = x + ⎪ ⎨ 12 0 1. x ⎪

1 ⎪ 3 6 ⎪ x + (x − t)3 u(t) + (x − t)2 v(t) dt ⎩ v(x) = x − 30 0 ⎧ x ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎪ u(x) = 1 + x + ⎨ 0

x ⎪ 2 ⎪ ⎪ (u(t) − v(t)) dt ⎩ v(x) = 1 − x − 2x2 − x3 + 3 0 ⎧ x 1 5 ⎪ ⎪ ((x − t)u(t) − (x − t)v(t)) dt x + ⎪ u(x) = 1 + x3 − ⎨ 10 0 3. x ⎪ 1 ⎪ ⎪ (u(t) − v(t)) dt ⎩ v(x) = 1 − x3 − x4 + 2 0 ⎧ x

1 6 ⎪ 2 3 ⎪ (x − t)2 u(t) − (x − t)2 v(t) dt ⎪ ⎨ u(x) = x + x − 30 x + 0 4. x ⎪ 1 ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎩ v(x) = x2 − x3 − x4 + 6 0 ⎧ x ⎪ ⎪ (cos(x − t)u(t) + sin(x − t)v(t)) dt ⎪ ⎨ u(x) = cos x + 2 sin x − 1 + 2.

5.

0

⎪ ⎪ ⎪ ⎩ v(x) = sin x + x cos x − 1 + ⎧ ⎪ ⎪ u(x) = x − x sin x + ⎪ ⎨

6.

x

0 x

⎪ ⎪ ⎪ ⎩ v(x) = 2 − 2 cosh x +

0

(sin(x − t)u(t) + cos(x − t)v(t)) dt

(sin(x − t)u(t) + cos(x − t)v(t)) dt

(cosh(x − t)u(t) + sinh(x − t)v(t)) dt

0

x

(cos(x − t)u(t) + sin(x − t)v(t)) dt

0

⎪ ⎪ ⎪ ⎩ v(x) = cos x − sin x + ⎧ ⎪ ⎪ u(x) = 2 − cosh x + ⎪ ⎨

7.

x

x

0

(sinh(x − t)u(t) + cosh(x − t)v(t)) dt

⎧ x ⎪ ⎪ (cosh(x − t)u(t) + sinh(x − t)v(t)) dt ⎪ u(x) = ex − sinh x − x sinh x + ⎨ 0 8. x ⎪ ⎪ ⎪ (sinh(x − t)u(t) + cosh(x − t)v(t)) dt ⎩ v(x) = e−x − x cosh x + 0

⎧ x ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎪ ⎨ u(x) = cos x + 3 sin x − 2x + 0 9. x ⎪ ⎪ ⎪ ((x − t)u(t) − (x − t)v(t)) dt ⎩ v(x) = cos x + sin x − 2 + 0

10.3 Systems of Volterra Integral Equations of the First Kind ⎧ x 1 x ⎪ ⎪ (sin x + cos x) + ((x − t)u(t) + (x − t)v(t)) dt e u(x) = ⎪ ⎨ 2 0 10. x ⎪ ⎪ ⎪ (u(t) + v(t)) dt ⎩ v(x) = ex (cos x − sin x) +

323

0

⎧ x ⎪ ⎪ (v(t) + w(t)) dt u(x) = cos x − sin x − 1 + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ x ⎨ (u(t) + w(t)) dt v(x) = 3 cos x − sin x − 2 + 11. ⎪ 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (u(t) + v(t)) dt ⎩ w(x) = 2 cos x − 1 + 0

⎧ x 1 4 ⎪ 2 ⎪ x + + (u(t) + v(t) − w(t)) dt u(x) = 1 − x ⎪ ⎪ ⎪ 4 0 ⎪ ⎪ ⎪ x ⎨ 2 1 (v(t) + w(t) − u(t)) dt v(x) = 2x + x2 − x3 − x4 + 12. ⎪ 3 4 0 ⎪ ⎪ ⎪ ⎪ x ⎪ 1 ⎪ ⎪ (w(t) + u(t) − v(t)) dt ⎩ w(x) = −x + x2 + x3 − x4 + 4 0

10.3 Systems of Volterra Integral Equations of the First Kind The standard form of the systems of Volterra integral equations of the ﬁrst kind is given by x % 1 (x, t)v(t) + · · · dt, f1 (x) = K1 (x, t)u(t) + K 0

f2 (x) = .. .

0

x

% 2 (x, t)v(t) + · · · dt, K2 (x, t)u(t) + K

(10.62)

% i (x, t), and the functions fi (x) are given where the kernels Ki (x, t) and K real-valued functions, and u(x), v(x), . . . are the unknown functions that will be determined. Recall that the unknown functions appear inside the integral sign for the Volterra integral equations of the ﬁrst kind. In this section we will discuss two main methods that are commonly used for handling the Volterra integral equations of the ﬁrst kind. Other methods are available in the literature but will not be presented in this text.

10.3.1 The Laplace Transform Method We ﬁrst begin by using the Laplace transform method. Recall that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by

324

10 Systems of Volterra Integral Equations

L{(f1 ∗ f2 )(x)} = L

0

x

f1 (x − t)f2 (t)dt

= F1 (s)F2 (s).

(10.63)

proceeding as in the previous section, we will examine speciﬁc systems of Volterra integral equations where the kernel is a diﬀerence kernel. We will apply the Laplace transform method and the inverse of the Laplace transform using Table 1.1 in Section 1.5. The Laplace transform method for solving systems of Volterra integral equations will be illustrated by studying the following examples. Example 10.9 Solve the system of Volterra integral equations of the ﬁrst kind by using the Laplace transform method x 1 2 1 3 1 4 x + x + x = ((x − t − 1)u(t) + (x − t + 1)v(t)) dt, 2 2 12 0 (10.64) x 1 4 3 2 1 3 x − x + x = ((x − t + 1)u(t) + (x − t − 1)v(t)) dt. 2 6 12 0 Taking Laplace transform of both sides of each equation in (10.64) gives 1 1 2 1 3 x + x + x4 = L {(x − t − 1) ∗ u(x) + (x − t + 1) ∗ v(x)} , L 2 2 12 1 3 2 1 3 x − x + x4 = L {(x − t + 1) ∗ u(x) + (x − t − 1) ∗ v(x)} . L 2 6 12 (10.65) This in turn gives 1 1 1 1 3 2 1 U (s) + V (s) = 3 + 4 + 5 , − + s2 s s2 s s s s (10.66) 1 1 1 2 1 1 3 + − − + U (s) + V (s) = s2 s s2 s s3 s4 s5 Solving this system of equations for U (s) and V (s) gives 1 2 1 1 U (s) = + 2 , V (s) = + 3 . (10.67) s s s s By taking the inverse Laplace transform of both sides of each equation in (10.67), the exact solutions are given by (10.68) (u(x), v(x)) = (1 + x, 1 + x2 ). Example 10.10 Solve the system of Volterra integral equations of the ﬁrst kind by using the Laplace transform method x ((x − t)u(t) + (x − t + 1)v(t)) dt, ex − 1 = 0 (10.69) x e−x − 1 = ((x − t − 1)u(t) + (x − t)v(t)) dt. 0

Taking Laplace transform of both sides of each equation in (10.69) gives

10.3 Systems of Volterra Integral Equations of the First Kind

325

L{ex − 1} = L{(x − t) ∗ u(x) + (x − t + 1) ∗ v(x)}, (10.70) L{e−x − 1} = L{(x − t − 1) ∗ u(x) + (x − t) ∗ v(x)}. This in turn gives 1 1 1 1 1 V (s) = − , U (s) + + 2 2 s s s s−1 s (10.71) 1 1 1 1 1 − V (s) = U (s) + − . s2 s s2 s+1 s Solving this system of equations for U (s) and V (s) gives 1 1 , V (s) = . (10.72) U (s) = s−1 s+1 By taking the inverse Laplace transform of both sides of each equation in (10.72), the exact solutions are given by (u(x), v(x)) = (ex , e−x ). (10.73) Example 10.11 Solve the system of Volterra integral equations of the ﬁrst kind by using the Laplace transform method x ((x − t − 1)u(t) − (x − t)v(t)) dt, 1 − sin x − cos x = 0 (10.74) x ((x − t)u(t) − (x − t − 1)v(t)) dt. 3 − sin x − 3 cos x = 0

Taking Laplace transform of both sides of each equation in (10.74) gives 1 1 1 1 s+1 − U (s) − 2 V (s) = − 2 , 2 s s s s s +1 (10.75) 1 1 3 3s + 1 1 U (s) − − V (s) = − . s2 s2 s s s2 + 1 Solving this system of equations for U (s) and V (s) gives s 1 s 1 + , V (s) = 2 − . (10.76) U (s) = 2 s + 1 s2 + 1 s + 1 s2 + 1 Taking the inverse Laplace transform of (10.76) gives the exact solutions by (u(x), v(x)) = (sin x + cos x, sin x − cos x). (10.77) Example 10.12 Solve the system of Volterra integral equations of the ﬁrst kind by using the Laplace transform method

326

10 Systems of Volterra Integral Equations

−2 + 2 cosh x = 1 −1 + x + x2 + ex = 2

x

(u(t) − v(t)) dt,

0 x

((x − t + 1)v(t) − (x − t − 1)w(t)) dt,

0

(10.78)

x 1 ((x − t)w(t) + (x − t + 1)u(t)) dt. x + x2 + xex = 2 0 Taking Laplace transform of both sides of each equation in (10.78) and solving the system of equations for U (s), V (s) and W (s) we ﬁnd 1 1 s 1 1 , V (s) = + , W (s) = U (s) = + (10.79) s s−1 s s+1 (s − 1)2 Taking the inverse Laplace transform of (10.79) gives the exact solutions by (u(x), v(x), w(x)) = (1 + ex , 1 + e−x , xex ).

(10.80)

Exercises 10.3.1 Use the Laplace transform method to solve the following systems of Volterra integral equations of the ﬁrst kind ⎧ x ⎪ ⎪ x2 + 2 x3 = ((x − t + 1)u(t) + (x − t − 1)v(t)) dt ⎪ ⎨ 3 0 1. x ⎪ 2 2 3 ⎪ ⎪ ((x − t − 1)u(t) + (x − t + 1)v(t)) dt ⎩x − x = 3 0 ⎧ x ⎪ 3 ⎪ ((x − t + 1)u(t) + (x − t − 1)v(t)) dt ⎪ ⎨x = 0

x ⎪ 1 ⎪ ⎪ ((x − t − 1)u(t) + (x − t + 1)v(t)) dt ⎩ − x3 = 3 0 ⎧ x 1 4 ⎪ 2 ⎪ 2x + = ((x − t + 1)u(t) + (x − t − 1)v(t)) dt x ⎪ ⎨ 2 0 3. x ⎪ 1 ⎪ ⎪ ((x − t − 1)u(t) + (x − t + 1)v(t)) dt ⎩ − x4 = 2 0 ⎧ x ⎪ ⎪ 2 + x − sin x − 2 cos x = ((x − t + 1)u(t) + (x − t)v(t)) dt ⎪ ⎨ 2.

4.

⎪ ⎪ ⎪ ⎩ 1 + x − cos x =

0

x 0

((x − t)u(t) + (x − t + 1)v(t)) dt

⎧ x 1 1 ⎪ ⎪ ((x − t + 1)u(t) + (x − t)v(t)) dt ⎪ x + x2 + x3 − sin x = ⎨ 2 3 0 5. x ⎪ ⎪ ⎪ ((x − t − 1)u(t) − (x − t + 1)v(t)) dt ⎩ x − x2 = 0

⎧ x ⎪ 2 ⎪ 2x + x + 2 sin x = ((x − t + 2)u(t) + (x − t − 2)v(t)) dt ⎪ ⎨ 0 6. x ⎪ ⎪ ⎪ ((x − t − 1)u(t) − (x − t + 1)v(t)) dt ⎩ −2x = 0

10.3 Systems of Volterra Integral Equations of the First Kind ⎧ x ⎪ 2 x ⎪ + 4e = ((x − t + 2)u(t) + (x − t − 2)v(t)) dt −4 + x ⎪ ⎨ 0 7. x ⎪ ⎪ ⎪ ((x − t + 1)u(t) − (x − t + 1)v(t)) dt ⎩ −2 − 4x + 2ex =

327

0

⎧ ⎪ x x ⎪ + 4xe = 6 − 6e ⎪ ⎨ 8.

0

⎪ ⎪ ⎪ ⎩ 4 − 4ex + 2xex =

⎧ ⎪ x ⎪ −2 − 2x + 2e = ⎪ ⎨ 9.

x

x

0 x

0

((x − t)u(t) − (x − t + 2)v(t)) dt ((x − t − 1)u(t) − (x − t + 1)v(t)) dt

((x − t − 1)u(t) + (x − t + 1)v(t)) dt

⎪ ⎪ ⎪ ⎩ −2 − 2x + 2x2 + 2ex =

x

0 x

((x − t + 1)u(t) + (x − t − 1)v(t)) dt

⎧ 1 3 1 4 ⎪ 2 ⎪ + + = ((x − t)u(t) + (x − t)v(t)) dt x x x ⎪ ⎪ ⎪ 6 12 0 ⎪ ⎪ ⎪ ⎨ x 1 1 ((x − t)v(t) + w(t)) dt 10. x + x2 + x4 = ⎪ 2 3 0 ⎪ ⎪ ⎪ x ⎪ ⎪ 1 3 ⎪ ⎪ ((x − t)w(t) + u(t)) dt x = ⎩ x + x2 + 20 0 ⎧ x 1 ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt x + x2 − sin x = ⎪ ⎪ ⎪ 2 0 ⎪ ⎪ ⎪ x ⎨ ((x − t)v(t) + w(t)) dt 11. x = ⎪ 0 ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ((x − t)w(t) + u(t)) dt ⎩ 1 + x − cos x = 0

⎧ x ⎪ ⎪ ((x − t)u(t) − (x − t)v(t) + w(t)) dt 2 − x − 2 cos x = ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ x ⎨ 1 (u(t) − (x − t)v(t) + w(t)) dt 12. 1 − x2 + sin x − cos x = ⎪ 2 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (u(t) − v(t) + (x − t)w(t)) dt ⎩ −2 + x + 2 cos x = 0

10.3.2 Conversion to a Volterra System of the Second Kind The conversion technique works eﬀectively by using Leibnitz rule that was presented in Section 1.3. Diﬀerentiating both sides of each equation in (10.62), and using Leibnitz rule, we obtain

328

10 Systems of Volterra Integral Equations

% 1 (x, x)v(x) f1 (x) = K1 (x, x)u(x) + K x % 1x (x, t)v(t) + · · · dt, + K1x (x, t)u(t) + K 0

f2 (x)

% 2 (x, x)v(x) = K2 (x, x)u(x) + K x % 2x (x, t)v(t) + · · · dt, + K2x (x, t)u(t) + K

(10.81)

0

.. . Three remarks can be made here: % i (x, x), i = 1, 2, . . . , n in each of the 1. If at least one of Ki (x, x) and K above equations does not vanish, then the system is reduced to a system of Volterra integral equations of the second kind. In this case, we can use any method that we studied before. % i (x, x) = 0, i = 1, 2, . . . , n, for any equation, and 2. If Ki (x, x) = 0 and K % if Kix (x, x) = 0 and Kix (x, x) = 0, then we diﬀerentiate again that equation. 3. The functions fi (x) must satisfy speciﬁc conditions to guarantee a unique continuous solution for each of the unknown solutions. The determination of these special conditions will be left as an exercise.

10.4 Systems of Volterra Integro-Diﬀerential Equations Volterra studied the hereditary inﬂuences when he was examining a population growth model. The research work resulted in a speciﬁc topic, where both diﬀerential and integral operators appeared together in the same equation. This new type of equations was termed as Volterra integro-diﬀerential equations, given in the form x u(i) (x) = f (x) + K(x, t)u(t)dt, (10.82) 0

n

where u(i) (x) = ddxui . Because the resulted equation combines the diﬀerential operator and the integral operator, then it is necessary to deﬁne initial conditions u(0), u (0), . . . , u(i−1) (0) for the determination of the particular solution u(x) of the Volterra integro-diﬀerential equation. The integro-diﬀerential equations were investigated in Chapter 5. In this section, we will study systems of Volterra integro-diﬀerential equations of the second kind given by x (i) % 1 (x, t)v(t) + · · · dt, u (x) = f1 (x) + K1 (x, t)u(t) + K 0 (10.83) x (i) % K2 (x, t)u(t) + K2 (x, t)v(t) + · · · dt. v (x) = f2 (x) + 0

10.4 Systems of Volterra Integro-Diﬀerential Equations

329

The unknown functions u(x), v(x), . . ., that will be determined, occur inside the integral sign whereas the derivatives of u(x), v(x), . . . appear mostly out% i (x, t), and the function side the integral sign. The kernels Ki (x, t) and K fi (x) are given real-valued functions. There is a variety of numerical and analytical methods that will be used for solving the system of integro-diﬀerential equations. However, in this section, we will present only two methods, new and traditional, that will be used for this study.

10.4.1 The Variational Iteration Method In Chapter 3, the variational iteration method (VIM) was used before in this text. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method [8] handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the so called Adomian polynomials that we need for nonlinear terms. The correction functionals for the Volterra system of integro-diﬀerential equations (10.83) are given by x t λ(t) u(i) (t) − f (t) − K(t, r)˜ u (r)dr dt, un+1 (x) = un (x) + 1 n n vn+1 (x) = vn (x) +

0 x

λ(t)

0

vn(i) (t)

− f2 (t) −

0 t

0

K(t, r)˜ vn (r)dr dt.

(10.84) As presented before, the variational iteration method is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ that can be identiﬁed optimally via integration by parts and by using a restricted variation. Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), n 0 and vn+1 (x), n 0 of the solutions u(x) and v(x). The zeroth approximations u0 (x) and v0 (x) can be any selective functions. However, the initial conditions are preferably used to select these approximations u0 and v0 (x) as will be seen later. Consequently, the solutions are given by u(x) = lim un (x), v(x) = lim vn (x). (10.85) n→∞

n→∞

The VIM will be illustrated by studying the following examples. Example 10.13 Use the VIM to solve the system of Volterra integro-diﬀerential equations

330

10 Systems of Volterra Integral Equations

1 1 u (x) = 1 + x − x2 + x3 + 2 3

x

0

3 1 v (x) = −1 − 3x − x2 − x3 + 2 3

((x − t)u(t) + (x − t + 1)v(t))dt,

x 0

((x − t + 1)u(t) + (x − t)v(t))dt,

(10.86) where u(0) = 1, v(0) = 1. The correction functionals for this system are given by un+1 (x) = un (x) x 1 2 1 3 un (t) − 1 − t + t − t − I1 (t) dt, − 2 3 0 x 3 2 1 3 vn (t) + 1 + 3t + t + t − I2 (t) dt, (10.87) vn+1 (x) = vn (x) − 2 3 0 where t

I1 (t) =

I2 (t) =

((t − r)un (r) + (t − r + 1)vn (r))dr,

0

0

(10.88)

t

((t − r + 1)un (r) + (t − r)vn (r))dr,

and λ = −1 for ﬁrst order integro-diﬀerential equation. We can use the initial conditions to select u0 (x) = u(0) = 1 and v0 (x) = v(0) = 1. Using this selection into the correction functionals gives the following successive approximations

u0 (x) = 1, v0 (x) = 1,

⎧ 1 3 1 4 ⎪ 2 ⎪ x , ⎨ u1 (x) = 1 + x + x + x + 6 12 ⎪ 1 1 4 ⎪ ⎩ v1 (x) = 1 − x − x2 − x3 − x , 6 12 ⎧ ⎪ u2 (x) = 1 + x + x2 + 1 x3 − 1 x3 + 1 x4 − 1 x4 − 1 x5 − 1 x6 , ⎪ ⎨ 6 6 12 12 120 360 ⎪ 1 1 1 1 1 1 6 ⎪ ⎩ v2 (x) = 1 − x − x2 + x3 − x3 + x4 − x4 + x5 + x , 6 6 12 12 120 360 ⎧ 1 6 1 5 1 5 1 6 ⎪ ⎪ u3 (x) = 1 + x + x2 + x − x + x − x + ··· , ⎨ 120 120 360 360 ⎪ 1 6 1 5 1 5 1 6 ⎪ ⎩ v3 (x) = 1 − x − x2 + + x − x x − x + ··· , 120 120 360 360

and so on. By canceling the noise terms, the exact solutions are given by (u(x), v(x)) = (1 + x + x2 , 1 − x − x2 ). (10.89) Example 10.14 Use the VIM to solve the system of Volterra integro-diﬀerential equations

10.4 Systems of Volterra Integro-Diﬀerential Equations

u (x) = 1 − x2 + ex + v (x) = 3 − 3ex +

0

x

(u(t) + v(t))dt, u(0) = 1, v(0) = −1, (10.90)

x

0

331

(u(t) − v(t))dt.

The correction functionals for this system are given by x t un (t) − 1 + t2 − et − un+1 (x) = un (x) − (un (r) + vn (r))dr dt, vn+1 (x) = vn (x) −

0 x

0

0

t (un (r) − vn (r))dr dt. vn (t) − 3 + 3et − 0

(10.91) We use the given initial conditions to select the zeroth approximations u0 (x) = u(0) = 1 and v0 (x) = v(0) = −1. Using this selection into the correction functionals gives the following successive approximations

u0 (x) = 1 v0 (x) = −1

x t ⎧ 2 t ⎪ ⎪ u (t) − 1 + t − e − (u u 1 (x) = u0 (x) − 0 (r) + v0 (r))dr dt ⎪ 0 ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ 1 3 ⎪ ⎨ = 1− x 3 ⎪ x t ⎪ ⎪ ⎪ t ⎪ v v (x) = v (x) − (t) − 3 + 3e − (u (r) − v (r))dr dt ⎪ 1 0 0 0 0 ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎩ = 2 + 3x − 3ex + x2 x t ⎧ 2 t ⎪ ⎪ u (x) = u (x) − (t) − 1 + t − e − (u (r) + v (r))dr dt u 2 1 1 1 1 ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ 1 3 1 4 1 2 ⎪ ⎪ + + + · · · = x + 1 + x + x x x ⎪ ⎨ 2! 3! 4! x t ⎪ ⎪ ⎪ ⎪ (u1 (r) − v1 (r))dr dt v2 (x) = v1 (x) − v1 (t) − 3 + 3et − ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ 1 2 1 3 1 4 1 ⎪ ⎩ = x − 1 + x + x + x + x + x5 + · · · 2! 3! 4! 5! ⎧ 1 1 1 1 ⎪ ⎪ u3 (x) = x + 1 + x + x2 + x3 + x4 + x5 + · · · ⎪ ⎨ 2! 3! 4! 5! ⎪ 1 1 1 1 1 ⎪ ⎪ ⎩ v3 (x) = x − 1 + x + x2 + x3 + x4 + x5 + x6 + · · · 2! 3! 4! 5! 6!

and so on. The exact solutions are therefore given by (u(x), v(x)) = (x + ex , x − ex ).

(10.92)

Example 10.15 Use the VIM to solve the system of Volterra integro-diﬀerential equations

332

10 Systems of Volterra Integral Equations

u (x) = −1 − x2 − sin x +

x

(u(t) + v(t))dt, u(0) = 1, u (0) = 1,

0

v (x) = 1 − 2 sin x − cos x +

x

0

(u(t) − v(t))dt, v(0) = 0, v (0) = 2.

(10.93) The correction functionals for this system are given by un+1 (x) = un (x) t x 2 (un (r) + vn (r))dr) dt, + (t − x)(un (t) + 1 + t + sin t − 0

0

vn+1 (x) = vn (x) x t (t − x)(vn (t) − 1 + 2 sin t + cos t − + (un (r) − vn (r))dr) dt, 0

0

(10.94) where we used λ = t − x for second-order integro-diﬀerential equation. We use the initial conditions to select u0 (x) = 1 + x and v0 (x) = 2x. Using this selection into the correction functionals gives the following successive approximations

u0 (x) = 1 + x,

v0 (x) = 2x, ⎧ 1 2 1 3 1 4 ⎪ ⎪ ⎨ u1 (x) = 1 − x + x + x + sin x, 2! 3! 4! ⎪ 1 1 1 ⎪ ⎩ v1 (x) = cos x + 2 sin x + x2 + x3 − x4 − 1, 2! 3! 4! ⎧ 1 1 1 ⎪ ⎪ u2 (x) = x + 1 − x2 + x4 − x6 + · · · , ⎨ 2! 4! 6! ⎪ 1 1 1 ⎪ ⎩ v2 (x) = x + x − x3 + x5 + x7 + · · · , 3! 5! 7!

and so on. The exact solutions are therefore given by (u(x), v(x)) = (x + cos x, x + sin x).

(10.95)

Example 10.16 Use the variational iteration method to solve the system of Volterra integrodiﬀerential equations x (6v(t) − 3w(t))dt, u(0) = 1, u (x) = 2 + ex − 3e2x + e3x + v (x) = ex + 2e2x − e3x +

2x

w (x) = −e + e x

+ 3e

3x

0

x

0

(3w(t) − u(t))dt, v(0) = 1,

+ 0

x

(u(t) − 2v(t))dt, w(0) = 1.

The correction functionals for this system are given by

(10.96)

10.4 Systems of Volterra Integro-Diﬀerential Equations

333

un+1 (x) = un (x) t x t 2t 3t (6vn (r) − 3wn (r)dr) dt, un (t) − 2 − e + 3e − e − − 0

0

vn+1 (x) = vn (x) t x (3wn (r) − un (r))dr dt, vn (t) − et − 2e2t + e3t − − 0

(10.97)

0

wn+1 (x) = wn (x) t x (un (r) − 2vn (r))dr dt. − wn (t) + et − e2t − 3e3t − 0

0

The initial conditions can be used to select the zeroth approximations as u0 (x) = 1, v0 (x) = 1 and w0 (x) = 1. Using this selection into the correction functionals gives the following successive approximations ⎧ ⎪ ⎨ u0 (x) = 1 v0 (x) = 1 ⎪ ⎩ w0 (x) = 1 ⎧ 7 3 1 3 ⎪ ⎪ u1 (x) = + 2x + x2 + ex − e2x + e3x ⎪ ⎪ 6 2 2 3 ⎪ ⎪ ⎨ 2 1 3x 2 x 2x v1 (x) = − + x + e + e − e ⎪ 3 3 ⎪ ⎪ ⎪ ⎪ 1 1 1 ⎪ ⎩ w1 (x) = − x2 − ex + e2x + e3x 2 2 2 ⎧ 1 1 ⎪ u2 (x) = 1 + x + x2 + x3 + 1 x4 + · · · ⎪ ⎪ ⎪ 2! 3! 4! ⎪ ⎪ ⎨ 1 1 1 2 v2 (x) = 1 + 2x + (2x) + (2x)3 + (2x)4 + · · · ⎪ 2! 3! 4! ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ w2 (x) = 1 + 3x + 1 (3x)2 + 1 (3x)3 + 1 (3x)4 + · · · 2! 3! 4!

and so on. The exact solutions are therefore given by (u(x), v(x), w(x)) = (ex , e2x , e3x ).

(10.98)

Exercises 10.4.1 Use the variational iteration method to solve the following systems of Volterra integrodiﬀerential equations ⎧ x 1 3 ⎪ ⎪ x u (x) = 2x + + ((1 − xt)u(t) − (1 + xt)v(t)) dt, u(0) = 1 ⎪ ⎨ 3 0 1. x ⎪ 1 ⎪ ⎪ ((1 + xt)u(t) + (1 − xt)v(t)) dt, v(0) = 1 ⎩ v (x) = −4x − x5 + 2 0

334 ⎧ ⎪ ⎪ u (x) ⎪ ⎪ ⎪ ⎨ 2. ⎪ ⎪ ⎪ v (x) ⎪ ⎪ ⎩ u(0) ⎧ ⎪ ⎪ ⎪ ⎨ u (x) 3.

10 Systems of Volterra Integral Equations = 3 − 3x +

3 3 x + 2

= −3 − 3x −

x 0

3 3 x + 2

= 1, v(0) = 2 = 1 − 4x + x3 +

⎪ ⎪ ⎪ ⎩ v (x) = −1 − x3 +

((1 − xt)u(t) + (1 − xt)v(t)) dt x

0

x

0

((1 + xt)u(t) + (1 + xt)v(t)) dt

((1 − xt)u(t) + (1 − xt)v(t)) dt, u(0) = 1

x

((1 + xt)u(t) + (1 + xt)v(t)) dt, v(0) = 1

0

x ⎧ ⎪ ⎪ ((1 − xt)u(t) + (1 − xt)v(t)) dt ⎪ u (x) = −2x + x3 + cos x + ⎪ ⎪ 0 ⎨ x 4. ⎪ ((1 + xt)u(t) + (1 + xt)v(t)) dt v (x) = −2x − x3 − cos x + ⎪ ⎪ ⎪ 0 ⎪ ⎩ u(0) = 1, v(0) = 1 ⎧ x ⎪ ⎪ (x) = 1 − x + sin x + ((x − t)u(t) − (x − t)v(t)) dt, u(0) = 1 u ⎪ ⎨ 5.

⎪ ⎪ ⎪ ⎩ v (x) = −1 + cos x +

0

x

0

(u(t) − v(t)) dt, v(0) = 2

x ⎧ ⎪ 2 ⎪ (x) = 1 − x − sin x + (u(t) + v(t)) dt u ⎪ ⎪ ⎪ 0 ⎨ x 6. 2 ⎪ ((1 − xt)u(t) − (1 + xt)v(t)) dt v (x) = 1 + x4 − sin x + ⎪ ⎪ ⎪ 3 0 ⎪ ⎩ u(0) = 1, v(0) = 1 x ⎧ ⎪ 2x ⎪ (x) = 2 − e + (u(t) + v(t)) dt u ⎪ ⎪ ⎪ 0 ⎪ ⎨ x 11 2x 3 7. + e (x) = − + ((x − t − 1)u(t) − (x − t + 1)v(t)) dt v ⎪ ⎪ ⎪ 2 2 0 ⎪ ⎪ ⎪ ⎩ u(0) = 1, v(0) = 2 ⎧ x ⎪ x ⎪ (x) = −3x + e + (u(t) + v(t)) dt, u(0) = 2 u ⎪ ⎨ 8.

0

⎪ ⎪ ⎪ ⎩ v (x) = 2 + x2 − 3ex +

x 0

(u(t) − (x − t)v(t)) dt, v(0) = 1

x ⎧ 2 4 ⎪ x ⎪ x u (x) = 3 + − e + ((1 − xt)u(t) − (1 + xt)v(t)) dt ⎪ ⎪ ⎪ 3 0 ⎨ x 9. 2 ⎪ ((1 + xt)u(t) − (1 − xt)v(t)) dt v (x) = 3 − x4 − 3ex + ⎪ ⎪ ⎪ 3 0 ⎪ ⎩ u(0) = 1, v(0) = −1 x ⎧ 1 2 ⎪ x ⎪ (x) = −x − + e + (v(t) + w(t)) dt, u(0) = 2 x u ⎪ ⎪ ⎪ 2 0 ⎪ ⎪ ⎪ x ⎨ 1 v (x) = x − x2 − ex + (w(t) − u(t)) dt, v(0) = 0 10. ⎪ 2 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (u(t) − v(t)) dt, w(0) = 1 ⎩ w (x) = 3 − ex + 0

10.4 Systems of Volterra Integro-Diﬀerential Equations ⎧ x 1 2 ⎪ ⎪ (x) = −x − − sin x + (v(t) + w(t)) dt, u(0) = 2 x u ⎪ ⎪ ⎪ 2 0 ⎪ ⎪ ⎪ x ⎨ 1 (w(t) − u(t)) dt, v(0) = 0 v (x) = x − x2 + sin x + 11. ⎪ 2 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (u(t) − v(t)) dt, w(0) = 1 ⎩ w (x) = 1 − 3 sin x +

335

0

⎧ x ⎪ ⎪ (u(t) − v(t) + w(t)) dt, u(0) = 2 u (x) = cos x − 2 sin x − ex + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ x ⎨ (u(t) − v(t) − w(t)) dt, v(0) = 1 v (x) = −2 − sin x + ex + 12. ⎪ 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (v(t) + w(t) − u(t)) dt, w(0) = 1 ⎩ w (x) = 2 + sin x − cos x + 0

10.4.2 The Laplace Transform Method The Laplace transform method was presented in Chapter 1, and was used in other chapters and in this chapter as well. Before we start applying this method, we summarize some of the concepts presented in this text. The Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x L{(f1 ∗ f2 )(x)} = L f1 (x − t)f2 (t)dt = F1 (s)F2 (s). (10.99) 0

Moreover, the Laplace Transforms of Derivatives can be summarized as follows: L{f (x)} = sL{f (x)} − f (0), L{f (x)} = s2 L{f (x)} − sf (0) − f (0), (10.100) 3 2 L{f (x)} = s L{f (x)} − s f (0) − sf (0) − f (0), and so on. Based on this summary, we will examine speciﬁc Volterra integrodiﬀerential equations where the kernel is a diﬀerence kernel. Recall that we will apply the Laplace transform method and the inverse of the Laplace transform using Table 1.1 in section 1.5. The Laplace transform method for solving systems of Volterra integro-diﬀerential equations will be illustrated by studying the following examples. Example 10.17 Solve the system of Volterra integro-diﬀerential equations by using the Laplace transform method x ((x − t)u(t) + (x − t)v(t)) dt, u(0) = 1 u (x) = 2x2 + 0 (10.101) x 1 v (x) = −3x2 − x5 + ((x − t)u(t) − (x − t)v(t)) dt, v(0) = 1. 10 0

336

10 Systems of Volterra Integral Equations

Notice that the kernels K1 (x − t) = K2 (x − t) = x − t. Taking Laplace transform of both sides of each equation gives 4 1 1 sU (s) − 1 = 3 + 2 U (s) + 2 V (s), s s s (10.102) 12 1 1 6 sV (s) − 1 = − 3 − 6 + 2 U (s) − 2 V (s), s s s s or equivalently 1 1 4 s − 2 U (s) − 2 V (s) = 1 + 3 , s s s (10.103) 6 12 1 1 s + 2 V (s) − 2 U (s) = 1 − 3 − 6 . s s s s Solving this system of equations for U (s) and V (s) gives 1 1 3! 3! (10.104) U (s) = + 4 , V (s) = − 4 . s s s s By taking the inverse Laplace transform of both sides of each equation, the exact solutions are given by (u(x), v(x)) = (1 + x3 , 1 − x3 ). (10.105) Example 10.18 Solve the system of Volterra integro-diﬀerential equations by using the Laplace transform method x (cos(x − t)u(t) + cos(x − t)v(t)) dt, u(0) = 1 u (x) = 1 − 2 sin x + 0

v (x) = −3 + 2 cos x +

0

x

(sin(x − t)u(t) + sin(x − t)v(t)) dt, v(0) = 1.

(10.106) Taking Laplace transform of both sides of each equation gives 1 2 s s sU (s) − 1 = − 2 + U (s) + 2 V (s), s s + 1 s2 + 1 s +1 (10.107) 3 2s 1 1 sV (s) − 1 = − + 2 + 2 U (s) + 2 V (s). s s +1 s +1 s +1 Solving this system of equations for U (s) and V (s) gives 1 1 1 1 (10.108) U (s) = + 2 , V (s) = − 2 . s s s s By taking the inverse Laplace transform of both sides of each equation, the exact solutions are given by (u(x), v(x)) = (1 + x, 1 − x). (10.109) Example 10.19 Solve the system of Volterra integro-diﬀerential equations by using the Laplace transform method

10.4 Systems of Volterra Integro-Diﬀerential Equations

u (x) = cos x − 2 sin x + v (x) = cos x + x cos x +

337

x

(cos(x − t)u(t) + sin(x − t)v(t)) dt,

0

x

(sin(x − t)u(t) + sin(x − t)v(t)) dt,

0

(10.110)

u(0) = 1, v(0) = −1. Taking Laplace transform of both sides of each equation gives s 2 s 1 sU (s) − 1 = 2 − + U (s) + 2 V (s), s + 1 s2 + 1 s2 + 1 s +1 (10.111) 1 s2 − 1 1 s + + 2 U (s) + V (s). sV (s) + 1 = 2 s + 1 (s + 1)2 s2 + 1 s2 + 1 Solving this system of equations for U (s) and V (s) gives s 1 s 1 (10.112) + , V (s) = 2 − . U (s) = 2 s + 1 s2 + 1 s + 1 s2 + 1 By taking the inverse Laplace transform of both sides of each equation, the exact solutions are given by (u(x), v(x)) = (sin x + cos x, sin x − cos x).

(10.113)

Example 10.20 Solve the system of Volterra integro-diﬀerential equations by using the Laplace transform method x u (x) = ex − e2x + e4x + (2v(t) − 4w(t)) dt, u(0) = 1, u (0) = 1, 0

v (x) = ex + 4e2x − e4x +

2x

w (x) = −e + e x

+ 16e

4x

x

0

(4w(t) − u(t)) dt, v(0) = 1, v (0) = 2,

+ 0

x

(u(t) − 2v(t)) dt, w(0) = 1, w (0) = 4.

(10.114) Taking Laplace transform of both sides of each equation gives 1 1 1 2 4 s2 U (s) − s − 1 = − + + V (s) − W (s), s−1 s−2 s−4 s s 1 4 1 4 1 (10.115) + − + W (s) − U (s), s2 V (s) − s − 2 = s−1 s−2 s−4 s s 1 1 16 1 2 s2 W (s) − s − 4 = − + + + U (s) − V (s). s−1 s−2 s−4 s s Solving this system of equations for U (s) and V (s) gives 1 1 1 (10.116) , V (s) = , W (s) = . U (s) = s−1 s−2 s−4 By taking the inverse Laplace transform of both sides of each equation, the exact solutions are given by (u(x), v(x), w(x)) = (ex , e2x , e4x ). (10.117)

338

10 Systems of Volterra Integral Equations

Exercises 10.4.2 Use the Laplace transform method to solve the following systems of Volterra integrodiﬀerential equations ⎧ x 1 3 ⎪ (x) = −x2 + ⎪ x u + ((x − t)u(t) + (x − t + 1)v(t)) dt, u(0) = 1 ⎪ ⎨ 6 0 1. x ⎪ 1 4 ⎪ ⎪ ((x − t)u(t) − (x − t)v(t)) dt, v(0) = 1 x + ⎩ v (x) = −x − 12 0 ⎧ x 1 ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt, u(0) = 0 ⎪ u (x) = 1 + x − x3 + ⎨ 3 0 2. x ⎪ 1 4 ⎪ ⎪ x + ((x − t)u(t) − (x − t)v(t)) dt, v(0) = 0 ⎩ v (x) = 1 − x − 12 0 x ⎧ ⎪ ⎪ (cos(x − t)u(t) + cos(x − t)v(t)) dt u (x) = 2 − 2 sin x + ⎪ ⎪ ⎪ 0 ⎨ x 3. ⎪ (x) = −4 + 2 cos x + (sin(x − t)u(t) + sin(x − t)v(t)) dt v ⎪ ⎪ ⎪ 0 ⎪ ⎩ u(0) = 1, v(0) = 1 ⎧ x

x−t ⎪ x ⎪ (x) = 3 + 3x − 2e + u(t) + ex−t v(t) dt, u(0) = 0 u e ⎪ ⎨ 4.

0

⎪ ⎪ ⎪ ⎩ v (x) = 3 + x + x2 − 2ex + ⎧ ⎪ ⎪ ⎪ u (x) = 2 cos x − ex + ⎨

5.

x

0

⎪ ⎪ ⎪ ⎩ v (x) = 1 − x − cos x +

x

0

x 0

x−t u(t) − ex−t v(t) dt, v(0) = 0 e

ex−t u(t) + ex−t v(t) dt, u(0) = 0 ((x − t)u(t) − (x − t)v(t)) dt, v(0) = 1

x ⎧ ⎪ ⎪ (cos(x − t)u(t) + sin(x − t)v(t)) dt u (x) = cos x − x sin x + ⎪ ⎪ ⎪ 0 ⎨ x 6. ⎪ (x) = −2 sin x + (sin(x − t)u(t) + cos(x − t)v(t)) dt v ⎪ ⎪ ⎪ 0 ⎪ ⎩ u(0) = 0, v(0) = 1 ⎧ x 3 2 ⎪ ⎪ (x) = −2x − + cos x + (u(t) + (x − t)v(t)) dt, u(0) = 2 u x ⎪ ⎨ 2 0 7. x ⎪ ⎪ ⎪ ((x − t)u(t) + v(t)) dt, v(0) = 2 ⎩ v (x) = −4x − x2 + 3 sin x + ⎧ ⎪ x ⎪ u (x) = 5 − 4e + ⎪ ⎨ 8.

0

x

0

⎪ ⎪ ⎪ ⎩ v (x) = −1 − 2xex +

ex−t u(t) + ex−t v(t) dt, u(0) = 3 x

0

⎧ ⎪ ⎪ ⎪ u (x) = 3 + 2x − 3ex + ⎨ 9.

x−t

e u(t) − ex−t v(t) dt, v(0) = 2

x−t u(t) + ex−t v(t) dt, u(0) = −1 e

x

0

⎪ ⎪ ⎪ ⎩ v (x) = 1 + ex + 2xex +

x

0

x−t u(t) − ex−t v(t) dt, v(0) = 1 e

References ⎧ x 1 2 ⎪ ⎪ (x) = −x − − sin x + (v(t) + w(t)) dt, u(0) = 2 x u ⎪ ⎪ ⎪ 2 0 ⎪ ⎪ ⎪ x ⎨ 1 (w(t) − u(t)) dt, v(0) = 0 v (x) = x − x2 + sin x + 10. ⎪ 2 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (u(t) − v(t)) dt, w(0) = 1 ⎩ w (x) = 1 − 3 sin x +

339

0

⎧ x ⎪ ⎪ (u(t) − v(t) + w(t)) dt, u(0) = 2 u (x) = cos x − 2 sin x − ex + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ x ⎨ (u(t) − v(t) − w(t)) dt, v(0) = 1 v (x) = −2 − sin x + ex + 11. ⎪ 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (v(t) + w(t) − u(t)) dt, w(0) = 1 ⎩ w (x) = 2 + sin x − cos x + 0

⎧ x ⎪ (x) = −x3 − x4 + ⎪ (3v(t) + 4w(t)) dt, u(0) = 0, u (0) = 1 u ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ x ⎨ (4w(t) − 2u(t)) dt, v(0) = 0, v (0) = 0 v (x) = 2 + x2 − x4 + 12. ⎪ 0 ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ (2u(t) − 3v(t)) dt, w(0) = 0, w (0) = 0 ⎩ w (x) = 6x − x2 + x3 + 0

References 1. A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 2. P. Linz, A simple approximation method for solving Volterra integro-diﬀerential equations of the ﬁrst kind, J. Inst. Maths. Appl., 14 (1974) 211–215. 3. P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, (1985). 4. D. Porter and D.S. Stirling, Integral Equations: a Practical Treatment from Spectral Theory to Applications, Cambridge, (2004). 5. G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994). 6. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 7. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 8. J.H. He, Variational iteration method for autonomous ordinary diﬀerential systems, Appl. Math. Comput., 114(2/3) (2000) 115–123.

Chapter 11

Systems of Fredholm Integral Equations

11.1 Introduction Systems of Volterra and Fredholm integral equations have attracted much concern in applied sciences. The systems of Fredholm integral equations appear in two kinds. The system of Fredholm integral equations of the ﬁrst kind [1–5] reads b ˜ 1 (x, t)v(t) dt, f1 (x) = K1 (x, t)u(t) + K a (11.1) b ˜ K2 (x, t)u(t) + K2 (x, t)v(t) dt, f2 (x) = a

where the unknown functions u(x) and v(x) appear only under the integral sign, and a and b are constants. However, for systems of Fredholm integral equations of the second kind, the unknown functions u(x) and v(x) appear inside and outside the integral sign. The second kind is represented by the form b ˜ 1 (x, t)v(t) dt, u(x) = f1 (x) + K1 (x, t)u(t) + K a

v(x) = f2 (x) +

b

˜ 2 (x, t)v(t) dt. K2 (x, t)u(t) + K

(11.2)

a

The systems of Fredholm integro-diﬀerential equations have also attracted a considerable size of interest. These systems are given by b ˜ 1 (x, t)v(t) dt, K1 (x, t)u(t) + K u(i) (x) = f1 (x) + a (11.3) b (i) ˜ K2 (x, t)u(t) + K2 (x, t)v(t) dt, v (x) = f2 (x) + a

where the initial conditions for the last system should be prescribed.

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

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11 Systems of Fredholm Integral Equations

11.2 Systems of Fredholm Integral Equations In this section we will study systems of Fredholm integral equations of the second kind given by b ˜ 1 (x, t)v(t) + · · · dt, u(x) = f1 (x) + K1 (x, t)u(t) + K a

v(x) = f2 (x) + .. .

b

˜ 2 (x, t)v(t) + · · · dt, K2 (x, t)u(t) + K

(11.4)

a

The unknown functions u(x), v(x), . . . that will be determined, appear inside ˜ i (x, t), and the funcand outside the integral sign. The kernels Ki (x, t) and K tion fi (x) are given real-valued functions. In what follows we will present the methods, new and traditional, that will be used. Recall that the Fredholm integral equations were presented in Chapter 4 where a variety of methods, new and traditional, were applied. In this section, we will focus our study only on two methods, namely the Adomian decomposition method and the direct computation method.

11.2.1 The Adomian Decomposition Method The Adomian decomposition method [6–7], as presented before, decomposes each solution as an inﬁnite sum of components, where these components are determined recurrently. This method can be used in its standard form, or combined with the noise terms phenomenon. Moreover, the modiﬁed decomposition method will be used wherever it is appropriate. Example 11.1 Use the Adomian decomposition method to solve the following system of Fredholm integral equations π u(x) = sin x − 2 − 2x − πx + ((1 + xt)u(t) + (1 − xt)v(t)) dt, 0 (11.5) π ((1 − xt)u(t) − (1 + xt)v(t)) dt. v(x) = cos x − 2 − 2x + πx + 0

The Adomian decomposition method suggests that the linear terms u(x) and v(x) be decomposed by an inﬁnite series of components ∞ ∞ un (x), v(x) = vn (x), u(x) = (11.6) n=0

n=0

where un (x) and vn (x), n 0 are the components of u(x) and v(x) that will be elegantly determined in a recursive manner.

11.2 Systems of Fredholm Integral Equations

343

Substituting (11.6) into (11.5) gives ∞ un (x) = sin x − 2 − 2x − πx n=0 π ∞ ∞ (1 + xt) un (t) + (1 − xt) vn (t) dt, + 0

∞ n=0

n=0

n=0

(11.7)

vn (x) = cos x − 2 − 2x + πx π ∞ ∞ un (t) − (1 + xt) vn (t) dt. + (1 − xt) 0

n=0

n=0

The modiﬁed decomposition method will be used here, hence we set the recursive relation u0 (x) = sin x − 2, v0 (x) = cos x − 2, π ((1 + xt)u0 (t) + (1 − xt)v0 (t)) dt, u1 (x) = −2x − πx + 0

v1 (x) = −2x + πx + uk+1 (x) = vk+1 (x) =

0

π

((1 − xt)u0 (t) − (1 + xt)v0 (t)) dt,

(11.8)

π 0

((1 − xt)uk (t) − (1 + xt)vk (t)) dt,

k 1,

((1 − xt)uk (t) − (1 + xt)vk (t)) dt,

k 1.

x 0

This in turn gives u0 (x) = sin x − 2, and v0 (x) = cos x − 2,

u1 (x) = 2 − 4π,

(11.9)

v1 (x) = 2 + 2π 2 x.

(11.10)

By canceling the noise terms ∓2 from u0 (x) and from v0 (x) we obtain the exact solutions (u(x), v(x)) = (sin x, cos x). (11.11) Example 11.2 Use the Adomian decomposition method to solve the following system of Fredholm integral equations π4 π3 2 + (tu(t) + tv(t)) dt, u(x) = x + sec x − 96 0 (11.12) π 4 π2 2 + (u(t) + v(t)) dt. v(x) = x − sec x − 16 0 Proceeding as before we obtain

344

11 Systems of Fredholm Integral Equations

∞

π3 + un (x) = x + sec2 x − 96 n=0 ∞

π2 + vn (x) = x − sec2 x − 16 n=0

π 4

0

0

π 4

t

∞

un (t) + t

n=0 ∞

un (t) +

n=0

∞

vn (t) dt,

n=0 ∞

(11.13)

vn (t) dt.

n=0

The modiﬁed decomposition method will be used here, hence we set the recursive relation u0 (x) = x + sec2 x, v0 (x) = x − sec2 x, π4 π3 u1 (x) = − (tu0 (t) + tv0 (t)) dt, + 96 0 π4 π2 (u0 (t) + v0 (t)) dt, v1 (x) = − + 16 (11.14) 0 π4 uk+1 (x) = (tuk (t) + tvk (t)) dt, k 1, vk+1 (x) =

0

π 4

0

(uk (t) + vk (t)) dt,

k 1.

This in turn gives and

u0 (x) = x + sec2 x,

u1 (x) = 0,

(11.15)

v0 (x) = x − sec2 x,

v1 (x) = 0.

(11.16)

As a result, the remaining components uk (x) and vk (x) for k 2 vanish. The exact solutions are given by (11.17) (u(x), v(x)) = (x + sec2 x, x − sec2 x). Example 11.3 Use the Adomian decomposition method to solve the following system of Fredholm integral equations 1 xt

2 x sinh(1 + x) + e u(t) + e−xt v(t) dt, u(x) = e − 1+x 0 (11.18) 1 −xt

2 xt −x sinh(1 − x) + e u(t) + e v(t) dt. v(x) = e − 1−x 0 Proceeding as before and using the modiﬁed decomposition method, we ﬁnd the recursive relation

11.2 Systems of Fredholm Integral Equations

v0 (x) = e−x ,

u0 (x) = ex ,

2 u1 (x) = sinh(1 + x) + 1+x v1 (x) = − uk+1 (x) = vk+1 (x) =

2 sinh(1 − x) + 1−x 1

1

ext u0 (t) + e−xt v0 (t) dt = 0,

0

0

1

e−xt u0 (t) + ext v0 (t) dt = 0,

ext uk (t) + e−xt vk (t) dt = 0,

0

0

1

345

e−xt uk (t) + ext vk (t) dt = 0,

(11.19)

k 1, k 1.

The exact solutions are given by (u(x), v(x)) = (ex , e−x ).

(11.20)

Example 11.4 Use the Adomian decomposition method to solve the following system of Fredholm integral equations 1 4 u(x) = x − + (v(t) + w(t)) dt, 3 −1 1 2 2 (11.21) (w(t) + u(t)) dt, v(x) = x + x − + 3 −1 1 2 w(x) = x2 + x3 − + (u(t) + v(t)) dt. 3 −1 Proceeding as before and using the modiﬁed decomposition method, we ﬁnd the recursive relation u0 (x) = x, v0 (x) = x + x2 , w0 (x) = x2 + x3 , 1 4 (v0 (t) + w0 (t)) dt = 0, u1 (x) = − + 3 −1 (11.22) 1 2 v1 (x) = − + (w0 (t) + u0 (t)) dt = 0, 3 −1 1 2 (u0 (t) + v0 (t)) dt = 0, w1 (x) = − + 3 −1 uk+1 (x) = 0, vk+1 = 0, wk+1 = 0, k 1. The exact solutions are given by (u(x), v(x), w(x)) = (x, x + x2 , x2 + x3 ).

(11.23)

346

11 Systems of Fredholm Integral Equations

Exercises 11.2.1 Solve the following systems of Fredholm integral equations 1 ⎧ 16 ⎪ ⎪ (tu(t) + tv(t)) dt + u(x) = x − ⎪ ⎨ 15 −1 1. 1 ⎪ 2 ⎪ ⎪ (xu(t) + xv(t)) dt ⎩ v(x) = x2 + x3 − x + 3 −1 1 ⎧ 2

2 2 2 ⎪ ⎪ x + (x − t2 )u(t) + (x2 − t2 )v(t) dt + u(x) = x − ⎪ ⎨ 3 5 −1 2. 1 ⎪

2 2 ⎪ ⎪ (x − t2 )u(t) − xtv(t) dt ⎩ v(x) = x2 + x3 + x + 5 −1 1 ⎧ 16 ⎪ 2 ⎪ − (u(t) + v(t)) dt u(x) = x + x + ⎪ ⎨ 15 −1 3. 1 ⎪ 16 2 ⎪ ⎪ ((x − t)u(t) + xtv(t)) dt x+ + ⎩ v(x) = x3 + x4 − 15 3 −1 ⎧ π ⎪ ⎪ (u(t) + v(t)) dt ⎪ ⎨ u(x) = sin x + cos x − 4 + 0 4. π ⎪ ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎩ v(x) = sin x − cos x − 4x + 2π + 0

π ⎧ ⎪ 2 2 ⎪ x − π + (u(t) + v(t)) dt u(x) = x + sin ⎪ ⎨ 0 5. π ⎪ π2 ⎪ ⎪ ⎩ v(x) = x − cos2 x − π 2 x + ((x − t)u(t) + (x + t)v(t)) dt + 2 0 π ⎧ ⎪ 2 ⎪ ((x − t)u(t) − (x + t)v(t)) dt ⎪ ⎨ u(x) = x + sin x + 2π + 6.

−π

⎪ ⎪ ⎪ ⎩ v(x) = x2 + cos x − 2π + ⎧ ⎪ ⎪ ⎪ ⎨ u(x) =

2−π 2

π

−π

((x + t)u(t) − (x − t)v(t)) dt

x + x tan−1 x +

1

−1

(xu(t) + xv(t)) dt

1 ⎪ 3π − 2 ⎪ ⎪ + x + tan−1 x + ⎩ v(x) = (tu(t) − tv(t)) dt 6 −1 ⎧ 1 ex ⎪ ⎪ u(x) = −x + + (xu(t) + xv(t)) dt ⎪ ⎨ 1 + ex 0 8. 1 ⎪ 1 ⎪ ⎪ ⎩ v(x) = −1 + + (u(t) + v(t)) dt 1 + ex 0 ⎧ π sin x 2 ⎪ ⎪ + u(x) = (2 − π)x + (xu(t) + xv(t)) dt ⎪ ⎨ 1 + sin x 0 9. π ⎪ cos x ⎪ 2 ⎪ ⎩ v(x) = (2 − π) + (u(t) + v(t)) dt + 1 + cos x 0

7.

11.2 Systems of Fredholm Integral Equations

347

√ ⎧ π 3− 3 6 ⎪ ⎪ + sec x tan x + u(x) = (u(t) − v(t)) dt ⎪ ⎨ 3 0 10. π ⎪ √ ⎪ 6 ⎪ ⎩ v(x) = (1 − 3)x + sec2 x + (xu(t) + xv(t)) dt 0

1 ⎧ 1 ⎪ ⎪ (v(t) − w(t)) dt + u(x) = x − ⎪ ⎪ 12 ⎪ −1 ⎪ ⎪ ⎪ 1 ⎨ 1 v(x) = x2 + + (w(t) − u(t)) dt 11. ⎪ 4 −1 ⎪ ⎪ ⎪ 1 ⎪ ⎪ 1 ⎪ ⎪ ⎩ w(x) = x2 − + (u(t) − v(t)) dt 6 −1 π ⎧ π 5π 4 ⎪ 2 ⎪ + sec x + (v(t) − w(t)) dt u(x) = 1 + ⎪ ⎪ 8 8 ⎪ 0 ⎪ ⎪ ⎪ π ⎨ π 3π 4 12. − sec2 x + v(x) = 1 − (w(t) − u(t)) dt ⎪ 8 8 ⎪ 0 ⎪ ⎪ ⎪ π ⎪ ⎪ π π ⎪ ⎩ w(x) = 1 − + sec2 x + 4 (u(t) − v(t)) dt 4 2 0

11.2.2 The Direct Computation Method It was stated before that all methods that we used in Chapter 4 to handle Fredholm integral equations can be used here to solve systems of Fredholm integral equations. In the previous section we selected the Adomian decomposition method. In this section we will employ the direct computational method. The other methods such as the variational iteration method, successive substitution method, and others can be used as well. The direct computational method will be applied to solve the systems of Fredholm integral equations of the second kind. The method was used before in this text, therefore we summarize the necessary steps. The method will be applied for the degenerate or separable kernels of the form n n gk (x)hk (t), K2 (x, t) = rk (x)sk (t). (11.24) K1 (x, t) = k=1

k=1

The direct computation method can be applied as follows: 1. We ﬁrst substitute (11.24) into the system of Fredholm integral equations the form b ˜ 1 (x, t)v(t) dt, K1 (x, t)u(t) + K u(x) = f1 (x) + a (11.25) b ˜ K2 (x, t)u(t) + K2 (x, t)v(t) dt. v(x) = f2 (x) + a

2. This substitution gives

348

11 Systems of Fredholm Integral Equations

u(x) = f1 (x) + v(x) = f2 (x) +

n k=1 n k=1

b

gk (x)

hk (t)u(t)dt +

a b

rk (x)

sk (t)u(t)dt + a

n k=1 n

b

g˜k (x)

(11.26)

b

r˜k (x)

k=1

˜ k (t)v(t)dt, h

a

s˜k (t)v(t)dt. a

3. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (11.26) becomes u(x) = f1 (x) + α1 g1 (x) + · · · + αn gn (x) + β1 g˜1 (x) + · · · + βn g˜n (x), (11.27) v(x) = f2 (x) + γ1 r1 (x) + · · · + γn rn (x) + δ1 r˜1 (x) + · · · + δn r˜n (x), where b

hi (t)u(t)dt, 1 i n,

αi = a

b

βi =

˜ i (t)v(t)dt, 1 i n, h

a

(11.28)

b

si (t)u(t)dt, 1 i n,

γi = a

b

s˜i (t)v(t)dt, 1 i n.

δi = a

4. Substituting (11.27) into (11.28) gives a system of n algebraic equations that can be solved to determine the constants αi , βi , γi , and δi . To facilitate the computational work, we can use the computer symbolic systems such as Maple and Mathematica. Using the obtained numerical values of these constants into (11.27), the solutions u(x) and v(x) of the system of Fredholm integral equations (11.25) follow immediately. Example 11.5 Solve the following system of Fredholm integral equations by using the direct computation method π u(x) = sin x + cos x − 4x + (xu(t) + xv(t))dt, 0 (11.29) π v(x) = sin x − cos x + (u(t) − v(t))dt. 0

Following the analysis presented above this system can be rewritten as u(x) = sin x + cos x + (α − 4)x, (11.30) v(x) = sin x − cos x + β, where π

π

(u(t) + v(t))dt,

α= 0

β= 0

(u(t) − v(t))dt.

To determine α, and β, we substitute (11.30) into (11.31) to obtain

(11.31)

11.2 Systems of Fredholm Integral Equations

349

π2 π2 α + πβ, β = −2π 2 + α − πβ. 2 2 Solving this system gives α = 4, β = 0. α = (4 − 2π2 ) +

Substituting (11.33) into (11.30) leads to the exact solutions (u(x), v(x)) = (sin x + cos x, sin x − cos x).

(11.32) (11.33) (11.34)

Example 11.6 Solve the following system of Fredholm integral equations by using the direct computation method π3 (tan tu(t) + sec tv(t))dt, u(x) = sec x − 2 + 0 (11.35) π3 √ (sec tu(t) + sec tv(t))dt. v(x) = −(1 + 3) + tan x + 0

Following the analysis presented above this system can be rewritten as u(x) = sec x − 2 + α1 + β1 , √ (11.36) v(x) = tan x − (1 + 3) + α2 + β1 , where π3 π3 π3 α1 = tan tu(t)dt, α2 = sec tu(t)dt, β1 = sec tv(t)dt. (11.37) 0

0

0

To determine α1 , α2 , and β1 , we substitute (11.36) into (11.37) and solve the resulting system we ﬁnd √ α1 = 1, α2 = 3, β1 = 1. (11.38) Substituting (11.38) into (11.36) leads to the exact solutions (u(x), v(x)) = (sec x, tan x).

(11.39)

Example 11.7 Solve the following system of Fredholm integral equations by using the direct computation method 1 (ln(xt)u(t) + ln(xt2 )v(t))dt, u(x) = 6 − ln x + 0+ (11.40) 1 v(x) = −4 + ln x +

0+

(ln(xt2 )u(t) − ln(xt)v(t))dt.

Proceeding as before, this system can be rewritten as u(x) = 6 + (α1 + β1 − 1) ln x + α2 + 2β2 , v(x) = −4 + (α1 − β1 + 1) ln x + 2α2 − β2 , where

(11.41)

350

11 Systems of Fredholm Integral Equations

α1 = β1 =

1

u(t)dt, 0+ 1

v(t)dt, 0+

α2 = β2 =

1

ln(t)u(t)dt, 0+ 1

(11.42) ln(t)v(t)dt.

0+

To determine αi , and βi , 1 i 2, we substitute (11.41) into (11.42) to obtain α2 = −8 + 2α1 − α2 + 2β1 − 2β2 , α1 = 7 − α1 + α2 − β1 + 2β2 , (11.43) β1 = −5 − α1 + 2α2 + β1 − β2 , β2 = 6 + 2α1 − 2α2 − 2β1 + β2 . Solving this system gives α1 = 0, α2 = 1, β1 = 2, β2 = −3. (11.44) These results lead to the exact solutions (u(x), v(x)) = (1 + ln x, 1 − ln x).

(11.45)

Example 11.8 Solve the following system of Fredholm integral equations by using the direct computation method π4 2 (v(t) − w(t))dt, u(x) = + sec2 x + 3 0 π4 10 (11.46) (w(t) − u(t))dt, v(x) = − sec2 x + 3 0 π4 4 w(x) = −1 − sec x + (u(t) − v(t))dt. 0

Proceeding as before, this system can be rewritten as 2 u(x) = + sec2 x + β − γ, 3 10 − sec2 x + γ − α, v(x) = 3 w(x) = −1 − sec4 x + α − β, where π4 π4 π4 u(t)dt, β = v(t)dt, γ = w(t)dt. α= 0

0

(11.47)

(11.48)

0

To determine α, β, and γ, we substitute (11.47) into (11.48) and by solving the resulting system we ﬁnd π π π 4 α = + 1, β = − 1, γ = − . (11.49) 4 4 4 3 These results lead to the exact solutions (11.50) (u(x), v(x), w(x)) = (1 + sec2 x, 1 − sec2 x, 1 − sec4 x).

11.2 Systems of Fredholm Integral Equations

351

Exercises 11.2.2 Use the direct computation method to solve the following systems of Fredholm integral equations 1 ⎧ 16 ⎪ ⎪ u(x) = x − (tu(t) + tv(t)) dt + ⎪ ⎨ 15 −1 1. 1 ⎪ 2 ⎪ ⎪ (xu(t) + xv(t)) dt ⎩ v(x) = x2 + x3 − x + 3 −1 ⎧ 1

⎪ ⎪ ln tu(t) − ln(xt2 )v(t) dt ⎪ ⎨ u(x) = −6 + 4 ln x + 2.

0+

⎪ ⎪ ⎪ ⎩ v(x) = −2 − 2 ln x +

1

0+

ln(xt2 )u(t) − ln tv(t) dt

1 ⎧ 16 ⎪ 2 ⎪ ⎪ ⎨ u(x) = x + x − 15 + −1 (u(t) + v(t)) dt 3. 1 ⎪ 2 16 ⎪ ⎪ x+ + ((x − t)u(t) + xtv(t)) dt ⎩ v(x) = x3 + x4 − 15 3 −1 ⎧ π ⎪ ⎪ (tu(t) + tv(t)) dt u(x) = sin x + cos x − 2π + ⎪ ⎨ 4.

0

⎪ ⎪ ⎪ ⎩ v(x) = sin x − cos x + πx +

π 0

(xu(t) − xtv(t)) dt

π ⎧ ⎪ 2 2 ⎪ x − π + (u(t) + v(t)) dt u(x) = x + sin ⎪ ⎨ 0 5. π ⎪ π2 ⎪ ⎪ ((x − t)u(t) + (x + t)v(t)) dt + ⎩ v(x) = x − cos2 x − π 2 x + 2 0 ⎧ π 3 ⎪ ⎪ u(x) = −2 + tan x + (sec tu(t) + tan tv(t)) dt ⎪ ⎨ 0

6.

π ⎪ π ⎪ 3 ⎪ ⎩ v(x) = − + sec x + (tan tu(t) − sec tv(t)) dt 3 0 π ⎧ ⎪ ⎪ u(x) = x2 + sin x + 2π + ((x − t)u(t) − (x + t)v(t)) dt ⎪ ⎨

7.

⎪ ⎪ ⎪ ⎩ v(x) = x2 + cos x − 2π +

−π

π

−π

((x + t)u(t) − (x − t)v(t)) dt

⎧ 1 ex ⎪ ⎪ + (xu(t) + xv(t)) dt u(x) = −x + ⎪ ⎨ 1 + ex 0 8. 1 ⎪ 1 ⎪ ⎪ ⎩ v(x) = −1 + + (u(t) + v(t)) dt 1 + ex 0 ⎧ π sin x 2 ⎪ ⎪ (xu(t) + xv(t)) dt u(x) = (2 − π)x + + ⎪ ⎨ 1 + sin x 0 9. π ⎪ cos x ⎪ 2 ⎪ ⎩ v(x) = (2 − π) + (u(t) + v(t)) dt + 1 + cos x 0

352

11 Systems of Fredholm Integral Equations

√ ⎧ π 3− 3 6 ⎪ ⎪ + sec x tan x + u(x) = (u(t) − v(t)) dt ⎪ ⎨ 3 0 10. π ⎪ √ ⎪ 6 ⎪ ⎩ v(x) = (1 − 3)x + sec2 x + (xu(t) + xv(t)) dt 0

1 ⎧ 1 ⎪ ⎪ (v(t) − w(t)) dt + u(x) = x − ⎪ ⎪ 12 ⎪ −1 ⎪ ⎪ ⎪ 1 ⎨ 1 v(x) = x2 + + (w(t) − u(t)) dt 11. ⎪ 4 −1 ⎪ ⎪ ⎪ 1 ⎪ ⎪ 1 ⎪ ⎪ ⎩ w(x) = x2 − + (u(t) − v(t)) dt 6 −1 π ⎧ √ π 4 ⎪ 2 ⎪ + x + 2 − (v(t) − w(t)) dt u(x) = 1 + sec ⎪ ⎪ 4 ⎪ 0 ⎪ ⎪ ⎪ π ⎨ √ π 4 12. v(x) = 1 − sec2 x − 2 − + (w(t) + u(t)) dt ⎪ 4 ⎪ 0 ⎪ ⎪ ⎪ π ⎪ ⎪ π ⎪ ⎩ w(x) = sec x tan x − + 4 (u(t) + v(t)) dt 2 0

11.3 Systems of Fredholm Integro-Diﬀerential Equations In Chapter 6, the Fredholm integro-diﬀerential equations were studied. The Fredholm integro-diﬀerential equations, were given in the form x K(x, t)u(t)dt, (11.51) u(i) (x) = f (x) + λ 0 i

where u(i) (x) = ddxui . Because the resulted equation combines the diﬀerential operator and the integral operator, then it is necessary to deﬁne initial conditions u(0), u (0), . . . , u(i−1) (0) for the determination of the particular solution u(x) of the Fredholm integro-diﬀerential equation (11.51). In this section, we will study systems of Fredholm integro-diﬀerential equations of the second kind given by b (i) ˜ 1 (x, t)v(t) dt, K1 (x, t)u(t) + K u (x) = f1 (x) + a (11.52) b ˜ 2 (x, t)v(t) dt. K2 (x, t)u(t) + K v (i) (x) = f2 (x) + a

The unknown functions u(x), v(x), . . ., that will be determined, occur inside the integral sign whereas the derivatives of u(x), v(x), . . . appear mostly out˜ i (x, t), and the function side the integral sign. The kernels Ki (x, t) and K fi (x) are given real-valued functions. In Chapter 6, four analytical methods were used for solving the Fredholm integro-diﬀerential equations. These methods are the variational iteration method (VIM), the Adomian decomposition method (ADM), the direct

11.3 Systems of Fredholm Integro-Diﬀerential Equations

353

computation method, and the series solution method. The aforementioned methods can eﬀectively handle the systems of Fredholm integro-diﬀerential equations (11.52). However, in this section, we will use only two of these methods, namely the direct computation method and the variational iteration method. The reader can apply the other two methods to show that it can be used to handle the system (11.52).

11.3.1 The Direct Computation Method The direct computation method is a reliable technique that was used before to handle Fredholm integral equations in Chapter 4, Fredholm integrodiﬀerential equations in Chapter 6, and systems of Fredholm integral equations in this chapter. The direct computation method will be applied to solve the systems of Fredholm integro-diﬀerential equations of the second kind in a parallel manner to our treatment that was used before. The method approaches any Fredholm equation in a direct manner and gives the solution in an exact form and not in a series form. The method will be applied for the degenerate or separable kernels of the form n n ˜ k (t), ˜ 1 (x, t) = gk (x)hk (t), K g˜k (x)h K1 (x, t) = K2 (x, t) =

k=1 n

rk (x)sk (t),

˜ 2 (x, t) = K

k=1

k=1 n

(11.53) r˜k (x)˜ sk (t).

k=1

The direct computation method can be applied as follows: 1. We ﬁrst substitute (11.53) into the system of Fredholm integro-diﬀerential equations (11.52) to obtain b b n n ˜hk (t)v(t)dt, gk (x) hk (t)u(t)dt + g˜k (x) u(i) (x) = f1 (x) + v (i) (x) = f2 (x) +

k=1 n k=1

a b

rk (x)

sk (t)u(t)dt + a

k=1 n k=1

a b

r˜k (x)

s˜k (t)v(t)dt. a

(11.54) 2. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (11.54) becomes u(i) (x) = f1 (x) + α1 g1 (x) + · · · + αn gn (x) + β1 g˜1 (x) + · · · + βn g˜n (x), v (i) (x) = f2 (x) + γ1 r1 (x) + · · · + γn rn (x) + δ1 r˜1 (x) + · · · + δn r˜n (x), (11.55) where

354

11 Systems of Fredholm Integral Equations

b

hi (t) u(t)dt, 1 i n,

αi = a

b

si (t) u(t)dt, 1 i n,

γi =

b

βi =

(11.56)

b

s˜i (t) v(t)dt, 1 i n.

δi =

a

˜hi (t) v(t)dt, 1 i n,

a

a

3. Integrating both sides of (11.55) i times from 0 to x, using the given initial conditions, and substituting the resulting equations for u(x) and v(x) into (11.56) gives a system of algebraic equations that can be solved to determine the constants αi , βi , γi , and δi . Using the obtained numerical values of these constants into the obtained equations for u(x) and v(x), the solutions u(x) and v(x) of the system of Fredholm integro-diﬀerential equations (11.52) follow immediately. Example 11.9 Solve the following system of Fredholm integro-diﬀerential equations by using the direct computation method π u (x) = sin x + x cos x + (2 − π2 ) + (tu(t) − v(t)) dt, u(0) = 0, 0 (11.57) π (u(t) − tv(t)) dt, v(0) = 0. v (x) = cos x − x sin x − 3π + 0

Following the analysis presented above, this system can be rewritten as u (x) = sin x + x cos x + (2 − π2 + α), (11.58) v (x) = cos x − x sin x + (β − 3π), where π

α= 0

π

(tu(t) − v(t))dt,

β= 0

(u(t) − tv(t))dt.

(11.59)

Integrating both sides of (11.58) once from 0 to x, and using the initial conditions we ﬁnd u(x) = x sin x + (2 − π 2 + α)x, (11.60) v(x) = x cos x + (β − 3π)x. To determine α, and β, we substitute (11.60) into (11.59) and solving the resulting system we obtain (11.61) α = π2 − 2, β = 3π. Substituting (11.61) into (11.60) leads to the exact solutions (u(x), v(x)) = (x sin x, x cos x).

(11.62)

Example 11.10 Solve the following system of Fredholm integro-diﬀerential equations by using the direct computation method

11.3 Systems of Fredholm Integro-Diﬀerential Equations

u (x) = −1 + sinh x +

ln 2

(u(t) + v(t))dt, u(0) = 1, 0

v (x) = 1 − 2 ln 2 + cosh x +

(11.63)

ln 2

(tu(t) + tv(t))dt, v(0) = 0. 0

This system can be rewritten as u (x) = sinh x + α − 1, v (x) = cosh x + 1 − 2 ln 2 + β, where ln 2 ln 2 (u(t) + v(t))dt, β = (tu(t) + tv(t))dt. α= 0

355

(11.64)

(11.65)

0

Integrating both sides of (11.64) once from 0 to x, and using the initial conditions we ﬁnd u(x) = cosh x + (α − 1)x, (11.66) v(x) = sinh x + (1 − 2 ln 2 + β)x. To determine α, and β, we substitute (11.66) into (11.65) and solving the resulting system we obtain α = 1, β = 2 ln 2 − 1. (11.67) Substituting (11.67) into (11.66) leads to the exact solutions (u(x), v(x)) = (cosh x, sinh x).

(11.68)

Example 11.11 Solve the following system of Fredholm integro-diﬀerential equations by using the direct computation method ln 2 5 x (u(t) + v(t))dt, u(0) = 1, u (x) = e − + 2 0 (11.69) ln 2 1 v (x) = 2e2x + + (tu(t) − tv(t))dt, v(0) = 1. 4 0 This system can be rewritten as 5 1 (11.70) u (x) = ex + α − , v (x) = 2e2x + + β, 2 4 where ln 2 ln 2 (11.71) α= (u(t) + v(t))dt, β = (tu(t) − tv(t))dt. 0

0

Integrating both sides of (11.70) once from 0 to x, and using the initial conditions we ﬁnd 5 1 x, v(x) = e2x + + β x. (11.72) u(x) = ex + α − 2 4 To determine α, and β, we substitute (11.72) into (11.71) and solving the resulting system we obtain

356

11 Systems of Fredholm Integral Equations

1 5 , β=− . 2 4 Substituting (11.73) into (11.72) leads to the exact solutions α=

(u(x), v(x)) = (ex , e2x ).

(11.73) (11.74)

Example 11.12 Solve the following system of Fredholm integro-diﬀerential equations by using the direct computation method π2 π ((x − t)u(t) − (x − t)v(t)) dt, u (x) = − cos x − 2 − + 2 0 u(0) = 1, u (0) = 0 π2 π ((x + t)u(t) − (x + t)v(t)) dt, v (x) = − sin x + 2 − + 2 0

(11.75)

v(0) = 0, v (0) = 1. This system can be rewritten as

π u (x) = − cos x − 2 − + αx − β, 2 π + αx + β, v (x) = − sin x + 2 − 2

where

π 2

α= 0

(u(t) − v(t))dt,

π 2

β= 0

(tu(t) − tv(t))dt.

(11.76)

(11.77)

Integrating both sides of (11.76) twice from 0 to x, and using the initial conditions and proceeding as before we obtain π α = 0, β = − 2. (11.78) 2 This in turn gives the exact solutions (u(x), v(x)) = (cos x, sin x).

(11.79)

Exercises 11.3.1 Use the direct computation method to solve the following systems of Fredholm integro-diﬀerential equations ⎧ π ⎪ ⎪ (u(t) − tv(t)) dt, u(0) = 0 ⎨ u (x) = cos x − 4 + 0 1. π ⎪ ⎪ (tu(t) − v(t)) dt, v(0) = 1 ⎩ v (x) = − sin x − π + 0

11.3 Systems of Fredholm Integro-Diﬀerential Equations π ⎧ π 2 ⎪ ⎪ (x) = − sin x − 2x + ((x − t)u(t) − (x − t)v(t)) dt + u ⎪ ⎪ 2 ⎪ 0 ⎪ ⎨ π π 2. 2 ⎪ ((x + t)u(t) − (x + t)v(t)) dt v (x) = − cos x − 2x − + ⎪ ⎪ 2 ⎪ 0 ⎪ ⎪ ⎩ u(0) = 2, v(0) = 1 π ⎧ π ⎪ ((x − t)u(t) − (x + t)v(t)) dt ⎪ u (x) = −2 sin(2x) − + ⎪ ⎪ 2 ⎪ 0 ⎨ π π 3. ⎪ ((x + t)u(t) − (x − t)v(t)) dt v (x) = 2 cos(2x) + + ⎪ ⎪ 2 ⎪ 0 ⎪ ⎩ u(0) = 1, v(0) = 0 ⎧ ln 2 1 ⎪ 2 ⎪ (ln 2) u (x) = sinh 2x + + (tu(t) − tv(t)) dt ⎪ ⎪ ⎪ 2 ⎪ 0 ⎨ ln 2 4. 15 ⎪ − 2 ln 2 + v (x) = sinh 2x − (u(t) + v(t)) dt ⎪ ⎪ ⎪ 16 0 ⎪ ⎪ ⎩ u(0) = 1, v(0) = 2 ⎧ ln 2 ⎪ 2 ⎪ (x) = sinh 2x − (ln 2) + (tu(t) + tv(t)) dt, u(0) = 2 u ⎪ ⎨ 5.

0

⎪ ⎪ ⎪ ⎩ v (x) = −2 sinh 2x − 2 ln 2 +

ln 2 0

(u(t) + v(t)) dt, v(0) = 0

⎧ ln 2 3 1 ⎪ ⎪ u (x) = cosh x + (tu(t) − tv(t)) dt, u(0) = 0 − ln 2 + ⎪ ⎨ 2 2 0 6. ln 2 ⎪ ⎪ ⎪ ⎩ v (x) = sinh x − (ln 2)2 + (u(t) + v(t)) dt, v(0) = 1 0

⎧ ln 2 2 ⎪ x 3 ⎪ (tu(t) + tv(t)) dt, u(0) = 1 ⎪ ⎨ u (x) = 1 + e − 3 (ln 2) + 0 7. ln 2 ⎪ ⎪ ⎪ ⎩ v (x) = 1 − ex − (ln 2)2 + (u(t) + v(t)) dt, v(0) = −1 0

ln 2 ⎧ 1 ⎪ x ⎪ (1 − 3 ln 2) + (x) = (1 + x)e + (u(t) + v(t)) dt u ⎪ ⎪ 2 ⎪ 0 ⎪ ⎨ ln 2 1 8. ⎪ v (x) = (1 − x)e−x + (3 − 5 ln 2) + (u(t) − v(t)) dt ⎪ ⎪ 2 ⎪ 0 ⎪ ⎪ ⎩ u(0) = 0, v(0) = 0 ⎧ ln 2 1 ⎪ x ⎪ + (x) = e − 2 ln 2 − (tu(t) + v(t)) dt, u(0) = 1 u ⎪ ⎨ 2 0 9. ln 2 ⎪ 1 ⎪ ⎪ ⎩ v (x) = 2e2x − 2 ln 2 − + (u(t) + tv(t)) dt, v(0) = 1 4 0

357

358

11 Systems of Fredholm Integral Equations ⎧ 4 ⎪ u (x) = 2 cos 2x + cos x ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ π ⎪ ⎪ ⎪ 2 ⎪ ⎪ + (sin(x − t)u(t) − cos(x − t)v(t)) dt, u(0) = 0, u (0) = 0 ⎨

10.

0

⎪ 2 ⎪ ⎪ ⎪ ⎪ v (x) = −2 cos 2x − cos x ⎪ 3 ⎪ ⎪ ⎪ π ⎪ ⎪ 2 ⎪ ⎩ + (cos(x − t)u(t) − sin(x − t)v(t)) dt, v(0) = 1, v (0) = 0 0

π ⎧ ⎪ (sin(x − t)u(t) − cos(x − t)v(t)) dt, u (x) = − sin x + π cos x + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎨ u(0) = 0, u (0) = 1 π 11. ⎪ ⎪ v (x) = − cos x − π sin x + ⎪ (cos(x − t)u(t) + sin(x − t)v(t)) dt, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩ v(0) = 1, v (0) = 0 ⎧ u (x) = −(1 + π) sin x − cos x ⎪ ⎪ π ⎪ ⎪ ⎪ ⎪ + (cos(x − t)u(t) + cos(x − t)v(t)) dt, u(0) = 1, u (0) = 1 ⎪ ⎨ 12.

0

⎪ v (x) = (1 − π) cos x − sin x ⎪ ⎪ ⎪ π ⎪ ⎪ ⎪ ⎩ (cos(x − t)u(t) − cos(x − t)v(t)) dt, v(0) = −1, v (0) = 1 + 0

11.3.2 The Variational Iteration Method In Chapters 4 and 6, the variational iteration method was used to handle the Fredholm integral equations and the Fredholm integro-diﬀerential equations respectively. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method [8] handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the so called Adomian polynomials that we need for nonlinear problems. The correction functionals for the system integro-diﬀerential equations (11.52) are given by un+1 (x) = un (x) b x (i) ˜ λ(t) un (t) − f1 (t) − un (r)dr + K1 (t, r)˜ vn (r)dr dt, + K1 (t, r)˜ 0

a

vn+1 (x) = vn (x) x λ(t) u(i) (t) − f (t) − + 2 n 0

a

b

(11.80) ˜ K2 (t, r)˜ un (r)dr + K2 (t, r)˜ vn (r)dr dt.

11.3 Systems of Fredholm Integro-Diﬀerential Equations

359

It is required ﬁrst to determine the Lagrange multiplier λ that can be identiﬁed optimally. Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), vn+1 (x), n 0 of the solutions u(x) and v(x). The zeroth approximations u0 (x) and v0 (x) can be any selective functions. However, we can use the initial conditions to select the zeroth approximations u0 and v0 (x). The VIM will be illustrated by studying the following examples. Example 11.13 Solve the system of Fredholm integro-diﬀerential equations by using the variational iteration method π

u (x) = −2 − sin x +

(u(t) + v(t))dt, u(0) = 1, 0

v (x) = 2 − π + cos x +

(11.81)

π

(tu(t) + tv(t))dt, v(0) = 0. 0

The correction functionals for this system are given by x un+1 (x) = un (x) − (un (t) + 2 + sin t − ρ1 ) dt, vn+1 (x) = vn (x) − where ρ1 =

0

0 x

0

(11.82) (vn (t)

+ π − 2 − cos t − ρ2 ) dt,

π

(un (r) + vn (r))dr,

ρ2 =

0

π

(run (r) + rvn (r))dr.

(11.83)

Selecting u0 (x) = 1 and v0 (x) = 0, the correction functionals gives the following successive approximations u0 (x) = 1, v0 (x) = 0, u1 (x) = u0 (x) −

0

x

u0 (t)

+ 2 + sin t −

0

π

(u0 (r) + v0 (r))dr dt,

= cos x − 2x + πx, x π (ru0 (r) + rv0 (r))dr dt, v0 (t) + π − 2 − cos t − v1 (x) = v0 (x) − 0

0

2

π = sin x + 2x − πx + x, (11.84) 2 x π (u1 (r) + v1 (r))dr dt, u1 (t) + 2 + sin t − u2 (x) = u1 (x) − 0

0

= cos x + (2x − 2x) + (πx − πx) + · · · , x π v1 (t) + π − 2 − cos t − v2 (x) = v1 (x) − (ru1 (r) + rv1 (r))dr dt, 0

0

360

11 Systems of Fredholm Integral Equations

= sin x + (2x − 2x) + (πx − πx) +

π2 π2 x− x + ··· , 2 2

and so on. It is obvious that noise terms appear in each component. By canceling the noise terms, the exact solutions are therefore given by (u(x), v(x)) = (cos x, sin x).

(11.85)

Example 11.14 Solve the system of Fredholm integro-diﬀerential equations by using the variational iteration method ln 2 (u(t) + v(t))dt, u(0) = 0, u (x) = −1 + cosh x + 0 (11.86) v (x) = 1 − 2 ln 2 + sinh x +

ln 2

(tu(t) + tv(t))dt, v(0) = 1. 0

The correction functionals for this system are given by x (un (t) + 1 − cosh t − ρ3 ) dt, un+1 (x) = un (x) − vn+1 (x) = vn (x) −

0 x

0

where

(11.87) (vn (t)

ρ3 = ρ4 =

+ 2 ln 2 − 1 − sinh t − ρ4 ) dt.

ln 2

(un (r) + vn (r))dr,

0

(11.88)

ln 2 0

(run (r) + rvn (r))dr.

We can use the initial conditions to select u0 (x) = 0 and v0 (x) = 1. Using this selection into the correction functionals gives the following successive approximations u0 (x) = 0, v0 (x) = 1, u1 (x) = sinh x + x ln 2 − x, v1 (x) = cosh x + x − 2x ln 2 +

x (ln 2)2 , 2

x

(11.89)

x (ln 2)2 + · · · , 2 2 x x v2 (x) = cosh x + (x − x) + (2x ln 2 − 2x ln 2) + (ln 2)2 − (ln 2)2 + · · · , 2 2 x x 3 3 u3 (x) = sinh x + (ln 2) − (ln 2) + · · · , 4 4 x x (ln 2)4 − (ln 2)4 + · · · , v3 (x) = cosh x + 3 3

u2 (x) = sinh x + (x − x) + (2x ln 2 − 2x ln 2) +

(ln 2)2 −

11.3 Systems of Fredholm Integro-Diﬀerential Equations

361

and so on. It is obvious that noise terms appear in each component. By canceling the noise terms, the exact solutions are therefore given by (u(x), v(x)) = (sinh x, cosh x). (11.90) Example 11.15 Solve the system of Fredholm integro-diﬀerential equations by using the variational iteration method ln 3 u (x) = ex − 6 + (u(t) + v(t))dt, u(0) = 1, 0 (11.91) ln 3

v (x) = 2e2x + 2 +

0

(u(t) − v(t))dt, v(0) = 1.

The correction functionals for this system are given by x ln 3 un (t) − et + 6 − (un (r) + vn (r))dr dt, un+1 (x) = un (x) − vn+1 (x) = vn (x) −

0

x 0

0

vn (t) − 2e2t − 2 −

0

ln 3

(un (r) − vn (r))dr

dt,

(11.92) where we used the Lagrange multiplier λ = −1 for the each of ﬁrst order Fredholm integro-diﬀerential equations. Selecting u0 (x) = 1 and v0 (x) = 1gives the following approximations u0 (x) = 1, v0 (x) = 1, u1 (x) = ex − 6x + 2x ln 3, v1 (x) = e2x + 2x, u2 (x) = ex + (6x − 6x) + (2x ln 3 − 2x ln 3) − 2x(ln 3)2 + x(ln 3)3 , (11.93) v2 (x) = e2x + (2x − 2x) − 4x(ln 3)2 + x(ln 3)3 , u3 (x) = ex + (2x(ln 3)2 − 2x(ln 3)2 ) + (x(ln 3)3 − x(ln 3)3 ) + · · · , v3 (x) = e2x + (4x(ln 3)2 − 4x(ln 3)2 ) + (x(ln 3)3 − x(ln 3)3 ) + · · · , and so on. It is obvious that noise terms appear in each component. By canceling the noise terms, the exact solutions are therefore given by (11.94) (u(x), v(x)) = (ex , e2x ). Example 11.16 Solve the system of Fredholm integro-diﬀerential equations by using the variational iteration method π 3π (2 + π) + u (x) = 2 cos 2x − (u(t) + tv(t))dt, u(0) = 1, u (0) = 0 4 0 π 3π (2 − π) + (tu(t) − v(t))dt, v(0) = 2, v (0) = 0. v (x) = −2 cos 2x + 4 0 (11.95)

362

11 Systems of Fredholm Integral Equations

The correction functionals for this system are given by x 3π un+1 (x) = un (x) + (t − x)(un (t) − 2 cos 2t + (2 + π) − ρ5 ) dt, 4 0 x 3π vn+1 (x) = vn (x) + (t − x)(vn (t) + 2 cos 2t − (2 − π) − ρ6 ) dt. 4 0 (11.96) where π

ρ5 =

0

π

(un (r) + rvn (r))dr,

ρ6 =

0

(run (r) − vn (r))dr.

(11.97)

Notice that the Lagrange multiplier λ = (t − x) because each equation is of second order. We can use the initial conditions to select u0 (x) = 1 and v0 (x) = 2. Using this selection into the correction functionals and proceeding as before we obtain the following successive approximations u0 (x) = 1, v0 (x) = 2, π u1 (x) = 1 + sin2 x − (2 − π)x2 , 8 π 2 (11.98) v1 (x) = 1 + cos x − (2 + π)x2 , 8 π π u2 (x) = 1 + sin2 x + (2 − π)x2 − (2 − π)x2 + · · · , 8 8 π π (2 + π)x2 − (2 + π)x2 + · · · , v2 (x) = 1 + cos2 x + 8 8 and so on. It is obvious that noise terms appear as explained in the previous examples. By canceling the noise terms, the exact solutions are therefore given by (u(x), v(x)) = (1 + sin2 x, 1 + cos2 x). (11.99)

Exercises 11.3.2 Use the variational iteration method to solve the following systems of Fredholm integro-diﬀerential equations ⎧ π ⎪ ⎪ (x) = cos x − 4 + (u(t) − tv(t)) dt, u(0) = 0 u ⎪ ⎨ 1.

0

⎪ ⎪ ⎪ ⎩ v (x) = − sin x − π +

π

0

(tu(t) − v(t)) dt, v(0) = 1

⎧ π π 2 ⎪ (x) = cos x − 2x + ⎪ + ((x − t)u(t) + (x − t)v(t)) dt, u(0) = 0 u ⎪ ⎨ 2 0 2. π ⎪ π ⎪ 2 ⎪ ⎩ v (x) = − sin x − 2 + + ((x + t)u(t) − (x + t)v(t)) dt, v(0) = 1 2 0

11.3 Systems of Fredholm Integro-Diﬀerential Equations ⎧ π ⎪ 2 ⎪ (x) = sin x + x cos x + (2 − π ) + (tu(t) − v(t)) dt, u(0) = 0 u ⎪ ⎨ 0 3. π ⎪ ⎪ ⎪ (u(t) − tv(t)) dt, v(0) = 0 ⎩ v (x) = cos x − x sin x − 3π + 0

⎧ ln 2 1 ⎪ 2 ⎪ (ln 2) (x) = sinh 2x + + (tu(t) − tv(t)) dt, u(0) = 1 u ⎪ ⎨ 2 0 4. ln 2 ⎪ 15 ⎪ ⎪ ⎩ v (x) = sinh 2x − − 2 ln 2 + (u(t) + v(t)) dt, v(0) = 2 16 0 ⎧ ln 2 ⎪ 2 ⎪ u (x) = sinh 2x − (ln 2) + (tu(t) + tv(t)) dt, u(0) = 1 ⎪ ⎨ 5.

0

⎪ ⎪ ⎪ ⎩ v (x) = − sinh 2x − 2 ln 2 +

ln 2

0

(u(t) + v(t)) dt, v(0) = 1

⎧ ln 2 3 1 ⎪ ⎪ (x) = cosh x + (tu(t) − tv(t)) dt, u(0) = 0 − ln 2 + u ⎪ ⎨ 2 2 0 6. ln 2 ⎪ ⎪ ⎪ ⎩ v (x) = sinh x − (ln 2)2 + (u(t) + v(t)) dt, v(0) = 1 0

⎧ ln 2 3 5 ⎪ x ⎪ u (x) = e + (tu(t) − tv(t)) dt, u(0) = 1 − ln 2 + ⎪ ⎨ 2 2 0 7. ln 2 ⎪ 3 ⎪ ⎪ ⎩ v (x) = −e−x − + (u(t) + v(t)) dt, v(0) = 1 2 0 ⎧ ln 2 1 ⎪ x ⎪ u (x) = (1 + x)e + (u(t) + v(t)) dt, u(0) = 0 (1 − 3 ln 2) + ⎪ ⎨ 2 0 8. ln 2 ⎪ 1 ⎪ ⎪ ⎩ v (x) = (1 − x)e−x + (3 − 5 ln 2) + (u(t) − v(t)) dt, v(0) = 0 2 0 ⎧ ln 2 10 ⎪ x ⎪ + (x) = e − (u(t) + v(t)) dt, u(0) = 1 u ⎪ ⎨ 3 0 9. ln 2 ⎪ 4 ⎪ ⎪ ⎩ v (x) = 3e3x + + (u(t) − v(t)) dt, v(0) = 1 3 0 ⎧ ln 2 10 ⎪ ⎪ + (u(t) + v(t)) dt, u(0) = 1, u (0) = 1 ⎪ u (x) = ex − ⎨ 3 0 10. ln 2 ⎪ 4 ⎪ ⎪ ⎩ v (x) = 9e3x + + (u(t) − v(t)) dt, v(0) = 1, v (0) = 3 3 0 π ⎧ ⎪ ⎪ (x) = − sin x + π cos x + (sin(x − t)u(t) − cos(x − t)v(t)) dt, u ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u(0) = 0, u (0) = 1 11. π ⎪ ⎪ ⎪ ⎪ (x) = − cos x − π sin x + (cos(x − t)u(t) + sin(x − t)v(t)) dt, v ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ v(0) = 1, v (0) = 0

363

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11 Systems of Fredholm Integral Equations

⎧ π ⎪ (cos(x − t)u(t) + cos(x − t)v(t)) dt, u (x) = −(1 + π) sin x − cos x + ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ u(0) = 1, u (0) = 1 π 12. ⎪ ⎪ ⎪ (cos(x − t)u(t) − cos(x − t)v(t)) dt, v (x) = (1 − π) cos x − sin x + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩ v(0) = −1, v (0) = 1

References 1. R.P. Agarwal, D. O’Regan and P. Wong, Eigenvalues of a system of Fredholm integral equations, Math. Comput. Modelling, 39 (2004) 1113–1150. 2. R.F. Churchhouse, Handbook of Applicable Mathematics, Wiley, New York, (1981). 3. R.P. Kanwal, Linear Integral Equations, Birkhauser, Boston, (1997). 4. K. Maleknejad, N. Aghazadeh and M. Rabbani, Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method, Appl. Math. Comput., 175 (2006) 1229-1234. 5. A. Polyanin and A. Manzhirov, Handbook of Integral Equations, Chapman and Hall, (2008). 6. G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994). 7. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 8. A.M. Wazwaz, The variational iteration method for solving linear and nonlinear systems of PDEs, Comput. Math. Appl., 54 (2007) 895–902.

Chapter 12

Systems of Singular Integral Equations

12.1 Introduction Systems of singular integral equations appear in many branches of scientiﬁc ﬁelds [1–6], such as microscopy, seismology, radio astronomy, electron emission, atomic scattering, radar ranging, plasma diagnostics, X-ray radiography, and optical ﬁber evaluation. Studies of systems of singular integral equations have attracted much concern in applied sciences. The use of computer symbolic systems such as Maple and Mathematica facilitates the tedious work of computation. The general ideas and the essential features of these systems are of wide applicability. The well known systems of singular integral equations [7] are given by x f1 (x) = (K11 (x, t)u(t) + K12 (x, t)v(t)) dt, 0 (12.1) x (K21 (x, t)u(t) + K22 (x, t)v(t)) dt. f2 (x) = 0

and u(x) = f1 (x) +

v(x) = f2 (x) +

x

(K11 (x, t)u(t) + K12 (x, t)v(t)) dt,

0

(12.2)

x 0

(K21 (x, t)u(t) + K22 (x, t)v(t)) dt,

where the kernels Kij are singular kernels given by 1 , 1 i, j 2. (12.3) Kij = (x − t)αij The system (12.1) and the system (12.2) are called the system of the generalized Abel singular integral equations and the system of the weakly generalized singular integral equations respectively. For αij = 12 , the system (12.1) is called the system of Abel singular integral equations.

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

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12 Systems of Singular Integral Equations

12.2 Systems of Generalized Abel Integral Equations In Chapter 7, the singular integral equations were presented. The Abel’s singular integral equation and its generalized form, and the weakly singular integral equations were handled by a variety of methods. In this section, the systems of Abel’s generalized singular integral equations will be examined by using only the Laplace transform method that we used before in Chapter 7 among other places. In this section, ﬁrst we will study systems of two unknown functions u(x) and v(x), and then we will study systems of three unknown functions u(x), v(x) and w(x) as well. Generalization to any number of unknowns can be easily developed.

12.2.1 Systems of Generalized Abel Integral Equations in Two Unknowns The system of generalized Abel integral equation in two unknowns is of the form x (K11 (x, t)u(t) + K12 (x, t)v(t)) dt, f1 (x) = 0 (12.4) x f2 (x) = (K21 (x, t)u(t) + K22 (x, t)v(t)) dt. 0

The kernels Kij (x, t), 1 i, j 2 and the functions fi (x), i = 1, 2 are given real-valued functions. Recall that the kernels Kij are singular kernels given by 1 , 1 i, j 2. (12.5) Kij = (x − t)αij For αij = 12 , 1 i, j 2, the system is called system of Abel integral equations. Abel’s systems of three equations in three unknowns will be examined in details in the next section. The system of Abel’s generalized singular integral equations (12.4) gives a solution if K11 K12 = 0. (12.6) det K21 K22 In what follows we will apply the Laplace transform method to handle the system (12.4). Taking Laplace transform of both sides of the system (12.4) gives the linear system in U (s) and V (s) F1 (s) = K11 (s)U (s) + K12 (s)V (s), (12.7) F2 (s) = K21 (s)U (s) + K22 (s)V (s), where

12.2 Systems of Generalized Abel Integral Equations

U (s) = L{u(x)}, Fi (s) = L{fi (x)}, Kij (s) = L{Kij (x)},

367

V (s) = L{v(x)}, 1 i 2, 1 i, j 2.

(12.8)

Solving (12.7) for U (s) and V (s) by using Cramer’s rule gives F1 (s) F2 (s)

K12 (s) K22 (s) , U (s) = K11 (s) K12 (s) K12 (s) K22 (s)

K11 (s) K21 (s)

F1 (s) F2 (s) . V (s) = K11 (s) K12 (s) K21 (s) K22 (s)

(12.9)

Having determined U (s) and V (s), the unique solution for u(x) and v(x) can be determined by using the inverse Laplace transform method. Table 1 in Section 1.5 shows the Laplace transforms and the inverse Laplace transforms for a variety of functions. It is interesting to point out that the computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations. This will be used here to facilitate the computational work. Moreover, recall that the linear system (12.7) has a unique solution for u(x) and v(x) if and only if K11 (s) K12 (s) (12.10) K12 (s) K22 (s) = 0. It is worth noting that for simplicity reasons we will focus our study only on the speciﬁc case where α11 = α22 , α12 = α21 . (12.11) The other cases of αij can be handled by a like manner. In this case, the use of the computer symbolic systems is necessary to perform tedious calculations. Example 12.1 Solve the system of singular integral equations by using the Laplace transform method x √ 3 2 2 1 3 1 3 dt, 2 x 1+ x + x 1− x = 1 u(t) + 1 v(t) 3 2 5 (x − t) 2 (x − t) 3 0 x √ 3 2 2 1 3 1 u(t) + v(t) dt. 2 x 1 − x + x3 1 + x = 1 1 3 2 5 (x − t) 3 (x − t) 2 0 (12.12) Taking Laplace transform of both sides of each equation in (12.12) gives √ −5 √ 2 2 −2 8 1 s− 3 (1 − s) = πs− 2 U (s) + Γ s 3 V (s), πs 2 (1 + s) − Γ 3 3 √ √ 2 2 5 8 2 1 s− 3 (1 + s) = Γ s− 3 U (s) + πs− 2 V (s). − πs− 2 (1 − s) + Γ 3 3 (12.13) Solving this systems of equations for U (s) and V (s), by using any computer symbolic system, such as Maple or Mathematica, gives 1 1 1 1 (12.14) U (s) = + 2 , V (s) = − 2 . s s s s

368

12 Systems of Singular Integral Equations

By taking the inverse Laplace transform of both sides of each equation in (12.14), the exact solutions are given by (u(x), v(x)) = (1 + x, 1 − x). (12.15) Example 12.2 Solve the system of singular integral equations by using the Laplace transform method 16 5 25 9 x 4 (32x2 + 39) + x 5 (57x2 − 133) 195 4788 x 1 1 dt, = 3 u(t) + 1 v(t) (x − t) 4 (x − t) 5 0 (12.16) 16 5 25 9 x 4 (32x2 − 39) + x 5 (57x2 + 133) 195 4788 x 1 1 u(t) + v(t) dt. = 1 3 (x − t) 5 (x − t) 4 0 Taking Laplace transform of both sides of each equation in (12.16) gives

√ √ Γ 45 Γ 45 π 2 π 2 6 6 + 1 + 14 − 1 = 1 3 U (s) + V (s),

9 4 s2 s2 s 4 Γ 34 s5 s4 Γ 4 s5

√ √ Γ 45 Γ 45 π 2 6 6 π 2 − 1 + 14 +1 = U (s) + 1 3 V (s).

9 4 s2 s2 s 4 Γ 34 s5 s5 s4 Γ 4 (12.17) Solving this systems of equations for U (s) and V (s), by using any computer symbolic system, such as Maple or Mathematica, gives 1 6 1 6 (12.18) U (s) = 4 + 2 , V (s) = 4 − 2 . s s s s By taking the inverse Laplace transform of both sides of each equation in (12.18), the exact solutions are given by (12.19) (u(x), v(x)) = (x3 + x, x3 − x).

Exercises 12.2.1 Solve the system of singular integral equations by using the Laplace transform method ⎧ x 5 4 1 1 8 1 ⎪ ⎪ ⎪ dt x 4 (6x + 5) + x 5 (20x + 27) = 3 u(t) + 1 v(t) ⎪ ⎨5 36 0 (x − t) 4 (x − t) 5 1. x ⎪ ⎪4 1 5 4 1 1 ⎪ ⎪ 4 5 x x (16x + 15) + (5x + 6) = u(t) + v(t) dt ⎩ 1 3 5 12 0 (x − t) 5 (x − t) 4

12.2 Systems of Generalized Abel Integral Equations 369 ⎧ x 1 5 4 3 2 1 ⎪ ⎪ 5 (5x + 9π) + 3 (−3x + 5π) = ⎪ x x dt 1 u(t) + 1 v(t) ⎪ ⎨ 36 10 0 (x − t) 5 (x − t) 3 2. x ⎪ ⎪ 3 2 1 1 5 4 ⎪ ⎪ 5 3 (−5x + 9π) + (3x + 5π) = u(t) + v(t) dt x x ⎩ 1 1 36 10 0 (x − t) 3 (x − t) 5 ⎧ x 5 4 1 4 1 1 ⎪ ⎪ ⎪ x 2 (x + 9) + x 5 (5x − 54) = dt 1 u(t) + 1 v(t) ⎪ ⎨3 36 0 (x − t) 2 (x − t) 5 3. x ⎪ ⎪ 1 5 4 1 4 1 ⎪ ⎪ 2 5 (x − 9) + (5x + 54) = u(t) + v(t) dt x x ⎩ 1 1 3 36 0 (x − t) 5 (x − t) 2 ⎧ x 1 256 9 1 16 3 ⎪ ⎪ 4 + 2 = ⎪ u(t) + v(t) dt x x 1 1 ⎪ ⎨ 3 315 0 (x − t) 2 (x − t) 4 4. x ⎪ ⎪ 1 8192 19 1 1 ⎪ ⎪ 4 2 x + 8x = u(t) + v(t) dt ⎩ 1 1 21945 0 (x − t) 4 (x − t) 2 ⎧ x 1 1 ⎪ ⎪ 2 x 12 (8x2 + 15) − 3 x 23 (9x2 − 20) = ⎪ u(t) + v(t) dt 1 1 ⎪ ⎨ 15 40 0 (x − t) 2 (x − t) 3 5. x ⎪ ⎪ 3 2 1 1 2 1 ⎪ 2 2 ⎪ 2 3 dt x (9x + 20) = ⎩ − x (8x − 15) + 1 u(t) + 1 v(t) 15 40 0 (x − t) 3 (x − t) 2 ⎧ x 1 1 ⎪ ⎪ 2 x 12 (16x3 + 35) + 3 x 23 (81x3 − 220) = ⎪ u(t) + v(t) dt 1 1 ⎪ ⎨ 35 440 0 (x − t) 2 (x − t) 3 6. x ⎪ ⎪ 1 2 1 3 2 1 ⎪ 3 3 ⎪ 2 3 x (16x − 35) + x (81x + 220) = dt ⎩ 1 u(t) + 1 v(t) 35 440 0 (x − t) 3 (x − t) 2 ⎧ 3 2 2 1 ⎪ x 3 (9x2 − 12x + 20) ⎪ − x 2 (8x2 − 10x − 15) + ⎪ ⎪ 15 40 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ dt = 1 u(t) + 1 v(t) ⎪ ⎨ 0 (x − t) 2 (x − t) 3 7. 2 1 3 2 ⎪ ⎪ ⎪ x 2 (8x2 − 10x + 15) − x 3 (9x2 − 12x − 20) ⎪ ⎪ 15 40 ⎪ ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ = u(t) + v(t) dt ⎩ 1 1 0 (x − t) 3 (x − t) 2 ⎧ 3 3 2 4 ⎪ x 4 (128x3 + 220x − 385) + x 3 (81x3 + 132x + 220) ⎪ − ⎪ ⎪ 1155 440 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ dt = 1 u(t) + 1 v(t) ⎪ ⎨ 0 (x − t) 3 (x − t) 4 8. 3 4 3 2 ⎪ ⎪ ⎪ x 4 (128x3 + 220x + 385) − x 3 (81x3 + 132x − 220) ⎪ ⎪ 1155 440 ⎪ ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ u(t) + v(t) dt = ⎩ 1 1 0 (x − t) 4 (x − t) 3

370

12 Systems of Singular Integral Equations

12.2.2 Systems of Generalized Abel Integral Equations in Three Unknowns The system of Abel’s generalized singular integral equations in three unknowns is of the form x

f1 (x) = f2 (x) = f3 (x) =

(K11 (x, t)u(t) + K12 (x, t)v(t) + K13 (x, t)w(t)) dt,

0 x

(K21 (x, t)u(t) + K22 (x, t)v(t) + K23 (x, t)w(t)) dt,

0

(12.20)

x 0

(K31 (x, t)u(t) + K32 (x, t)v(t) + K33 (x, t)w(t)) dt.

The kernels Kij (x, t), 1 i, j 3 and the functions fi (x), i = 1, 2, 3 are given real-valued functions. Recall that the kernels Kij are singular kernels given by 1 Kij = , 1 i, j 3. (12.21) (x − t)αij For αij = 12 , 1 i, j 3, the system is called system of Abel integral equations in three unknowns. The system of Abel’s generalized singular integral equations in three unknowns gives a solution if ⎤⎞ ⎛⎡ K11 K12 K13 (12.22) det ⎝⎣ K21 K22 K23 ⎦⎠ = 0. K31 K32 K33 In what follows we will apply the Laplace transform method to handle the system (12.20). Taking Laplace transform of both sides of the system (12.20) gives the linear system in U (s), V (s), and W (s) F1 (s) = K11 (s)U (s) + K12 (s)V (s) + K13 (s)W (s), F2 (s) = K21 (s)U (s) + K22 (s)V (s) + K23 (s)W (s), (12.23) F3 (s) = K31 (s)U (s) + K32 (s)V (s) + K33 (s)W (s). where U (s) = L{u(x)}, V (s) = L{v(x)}, W (s) = L{w(x)}, (12.24) Fi (s) = L{fi (x)}, 1 i 3, Kij (s) = L{Kij (x)}, 1 i, j 3, and the singular kernels Kij are given above in (12.21). Solving (12.23) for U (s), V (s), and W (s) by using Cramer’s rule gives F (s) K (s) K (s) 1 12 13 F (s) K (s) K (s) 22 23 2 F3 (s) K32 (s) K33 (s) , (12.25) U (s) = K11 (s) K12 (s) K13 (s) K (s) K (s) K (s) 22 23 21 K31 (s) K32 (s) K33 (s)

12.2 Systems of Generalized Abel Integral Equations

371

K (s) 11 K (s) 21 K31 (s)

K (s) 11 K (s) 21 K31 (s)

F1 (s) K13 (s) F2 (s) K23 (s) F3 (s) K33 (s) , V (s) = K11 (s) K12 (s) K13 (s) K (s) K (s) K (s) 22 23 21 K31 (s) K32 (s) K33 (s) and

K12 (s) F1 (s) K22 (s) F2 (s) K23 (s) F3 (s) . W (s) = K11 (s) K12 (s) K13 (s) K (s) K (s) K (s) 22 23 21 K31 (s) K32 (s) K33 (s)

(12.26)

(12.27)

The linear system (12.23) gives a unique solution for u(x), v(x) and w(x) if and only if K11 (s) K12 (s) K13 (s) K21 (s) K22 (s) K23 (s) = 0. (12.28) K31 (s) K32 (s) K33 (s) It is worth noting that for simplicity reasons we will focus our study only on the speciﬁc case where α11 = α22 , α12 = α21 . (12.29) Having determined U (s), V (s), and W (s), the unique solution for u(x), v(x), and w(x) can be determined by using the inverse Laplace transform method. It is interesting to point out that the computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations. This will be used here to facilitate the computational work. For simplicity reasons we will focus our study only on the speciﬁc case where For the ﬁrst equation in (12.20): α11 = α13 , v(x) = 0, For the second equation in (12.20): α22 = α23 , u(x) = 0, (12.30) For the third equation in (12.20): α31 = α32 , w(x) = 0. Example 12.3 Solve the system of singular integral equations by using the Laplace transform method x 1 1 2√ x(15 + 8x2 ) = u(t) + w(t) dt, 1 1 15 (x − t) 2 (x − t) 2 0 x 1 9 4 1 3 dt, x (7 + 6x) = (12.31) 2 v(t) + 2 w(t) 28 (x − t) 3 (x − t) 3 0 x 4 1 1 1 x 4 (5 + 4x) = dt. 3 u(t) + 3 v(t) 5 (x − t) 4 (x − t) 4 0

372

12 Systems of Singular Integral Equations

Taking Laplace transform of both sides of each equation in (12.31) gives * √ −3 π 2 (U (s) + W (s)), πs 2 1 + 2 = s s 2π 2 2π 7 1 √ 2 s− 3 1 + = √ 2 s− 3 (V (s) + W (s)), (12.32) s 3Γ√ 3 √3Γ 3 1 2π 2π 5 1 3 s− 4 1 + = 3 s− 4 (U (s) + V (s)) s Γ 4 Γ 4 Solving this systems of equations for U (s), V (s) and W (s) by using any computer symbolic system, such as Maple or Mathematica, gives 1 1 2 U (s) = , V (s) = 2 , W (s) = 3 . (12.33) s s s By taking the inverse Laplace transform of both sides of each equation in (12.33), the exact solutions are given by (12.34) (u(x), v(x), w(x)) = (1, x, x2 ). Example 12.4 Solve the system of singular integral equations by using the Laplace transform method x 1 1 1 u(t) + w(t) dt, 3x 3 (2 + 3x) = 2 2 (x − t) 3 (x − t) 3 0 x 1 8 1 1 2 4 dt, x (15 − 32x ) = (12.35) 3 v(t) + 3 w(t) 15 (x − t) 4 (x − t) 4 0 x 1 10 3 1 u(t) + v(t) dt. x5 = 2 2 3 (x − t) 5 (x − t) 5 0 Taking Laplace transform of both sides of each equation in (12.35) gives 4π 2 2π 4 1 − √ 2 s 3 1 + s− 3 (U (s) + W (s)), = √ 2 s 3Γ 3 3Γ( 3 ) √ √ 2π − 1 2 2π − 5 6 (12.36) s 4 1 − 2 = s 4 (V (s) + W (s)), s Γ 34 Γ( 34 ) 8 3 2Γ 35 s− 5 = Γ 35 s− 5 (U (s) + V (s)) Solving this systems of equations for U (s), V (s), and W (s) by using any computer symbolic system, such as Maple or Mathematica, gives 2 2 2 1 6 1 6 1 6 U (s) = + 2 + 3 , V (s) = − 2 − 3 , W (s) = + 2 − 3 . (12.37) s s s s s s s s s By taking the inverse Laplace transform of both sides of each equation in (12.37), the exact solutions are given by (u(x), v(x), w(x)) = (1 + 2x + 3x2 , 1 − 2x − 3x2 , 1 + 2x − 3x2 ).

(12.38)

12.2 Systems of Generalized Abel Integral Equations

373

Exercises 12.2.2 Solve the system of singular integral equations by using the Laplace transform method x ⎧ 1 4 3 ⎪ 2 (24x2 + 35) = ⎪ x 1 (u(t) + w(t)) dt ⎪ 105 ⎪ ⎪ 0 (x − t) 2 ⎪ ⎪ x ⎪ ⎨ 27 8 1 x 3 (9x + 11) = 1. 1 (v(t) + w(t)) dt 440 ⎪ 0 (x − t) 3 ⎪ ⎪ ⎪ ⎪ x ⎪ 16 7 1 ⎪ ⎪ x 4 (8x + 11) = ⎩ 1 (u(t) + v(t)) dt 231 0 (x − t) 4 x ⎧ 1 4 1 ⎪ ⎪ x 2 (8x + 9) = 1 (u(t) + w(t)) dt ⎪ ⎪ 3 ⎪ 0 (x − t) 2 ⎪ ⎪ ⎪ x ⎨ 3 2 1 x 3 (27x + 35) = 2. 1 (v(t) + w(t)) dt 10 ⎪ 0 (x − t) 3 ⎪ ⎪ x ⎪ ⎪ ⎪ 1 4 3 ⎪ ⎪ ⎩ x 4 (4x + 5) = 1 (u(t) + v(t)) dt 3 0 (x − t) 4 x ⎧ 1 3 2 ⎪ ⎪ x 3 (81x3 + 99x2 + 132x + 220) = 1 (u(t) + w(t)) dt ⎪ ⎪ 440 ⎪ 0 (x − t) 3 ⎪ ⎪ ⎪ x ⎨ 16 7 1 x 4 (32x2 + 80x + 55) = 3. 1 (v(t) + w(t)) dt 1155 ⎪ 0 (x − t) 4 ⎪ ⎪ x ⎪ ⎪ ⎪ 1 5 4 ⎪ ⎪ x 5 (25x2 + 70x + 63) = ⎩ 1 (u(t) + v(t)) dt 252 0 (x − t) 5 x ⎧ 1 3 1 ⎪ ⎪ x 3 (81x3 + 90x2 + 105x + 140) = 2 (u(t) + w(t)) dt ⎪ ⎪ 140 ⎪ 0 (x − t) 3 ⎪ ⎪ ⎪ x ⎨ 16 5 1 x 4 (96x2 + 208x + 117) = 4. 3 (v(t) + w(t)) dt 585 ⎪ 0 (x − t) 4 ⎪ ⎪ x ⎪ ⎪ ⎪ 1 5 3 ⎪ ⎪ x 5 (25x2 + 65x + 52) = ⎩ 2 (u(t) + v(t)) dt 156 0 (x − t) 5 x ⎧ 1 4 1 ⎪ 2 (8x3 − 70x − 105) = ⎪ x − 1 (u(t) + w(t)) dt ⎪ ⎪ 35 ⎪ 0 (x − t) 2 ⎪ ⎪ x ⎪ ⎨ 1 9 1 x 3 (27x3 − 30x2 − 560) = 5. − 2 (v(t) + w(t)) dt 140 ⎪ 0 (x − t) 3 ⎪ ⎪ ⎪ ⎪ x ⎪ 1 8 3 ⎪ ⎪ x 4 (16x2 + 132x + 231) = ⎩ 1 (u(t) + v(t)) dt 231 0 (x − t) 4 x ⎧ 1 9 2 ⎪ 3 (3x2 − 8x − 20) = ⎪ x − 1 (u(t) + w(t)) dt ⎪ 40 ⎪ ⎪ 0 (x − t) 3 ⎪ ⎪ x ⎪ ⎨ 1 5 3 x 5 (50x2 + 65x − 208) = 6. − 2 (v(t) + w(t)) dt 312 ⎪ 0 (x − t) 5 ⎪ ⎪ ⎪ ⎪ x ⎪ 1 4 1 ⎪ ⎪ x 4 (64x2 − 36x + 135) = ⎩ 3 (u(t) + v(t)) dt 45 0 (x − t) 4

374 ⎧ 9 5 ⎪ ⎪ x 3 (81x2 ⎪ ⎪ 220 ⎪ ⎪ ⎪ ⎪ ⎨ 25 8 x 5 (25x2 7. 468 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 32 5 ⎪ ⎪ x 4 (96x2 ⎩ 585 ⎧ 27 5 ⎪ ⎪ x 3 (45x2 ⎪ ⎪ 440 ⎪ ⎪ ⎪ ⎪ ⎨ 25 8 x 5 (25x2 8. 468 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 32 5 ⎪ ⎪ x 4 (96x2 ⎩ 585

12 Systems of Singular Integral Equations

x

1 1 (u(t) + w(t)) dt (x − t) 3 x 1 + 30x + 39) = 2 (v(t) + w(t)) dt 0 (x − t) 5 x 1 + 208x + 117) = 3 (u(t) + v(t)) dt 0 (x − t) 4 x 1 + 33x + 44) = 1 (u(t) + v(t) + w(t)) dt 0 (x − t) 3 x 1 + 30x + 39) = 2 (v(t) + w(t)) dt 0 (x − t) 5 x 1 + 208x + 117) = 3 (u(t) + v(t)) dt 0 (x − t) 4 + 44) =

0

12.3 Systems of the Weakly Singular Volterra Integral Equations In this section we will study systems of the weakly singular integral equations in two unknowns u(x) and v(x). Generalization to any number of unknowns can be followed in a parallel manner. Recall that the kernel is called weakly singular as the singularity may be transformed away by a change of variable [2]. In this section we will use only the Laplace transform method and the Adomian decomposition method to handle this type of systems.

12.3.1 The Laplace Transform Method The system of weakly singular Volterra integral equations of the convolution type in two unknowns is of the form x (K11 (x, t)u(t) + K12 (x, t)v(t)) dt, u(x) = f1 (x) + 0 (12.39) x (K21 (x, t)u(t) + K22 (x, t)v(t)) dt. v(x) = f2 (x) + 0

The kernels Kij (x, t), 1 i, j 2 and the functions fi (x), i = 1, 2 are given real-valued functions. The kernels Kij are singular kernels given by 1 Kij = , 1 i, j 2. (12.40) (x − t)αij Taking Laplace transforms of both sides of the system (12.39) gives the linear system in U (s) and V (s) U (s) = F1 (s) + K11 (s)U (s) + K12 (s)V (s), (12.41) V (s) = F2 (s) + K21 (s)U (s) + K22 (s)V (s),

12.3 Systems of the Weakly Singular Volterra Integral Equations

375

or equivalently (1 − K11 (s))U (s) − K12 (s)V (s) = F1 (s), −K21 (s)U (s) + (1 − K22 (s))V (s) = F2 (s),

(12.42)

where U (s) = L{u(x)}, V (s) = L{v(x)}, Fi (s) = L{fi (x)}, 1 i 2, Kij (s) = L{Kij (x)}, 1 i, j 2.

(12.43)

Solving (12.41) for U (s) and V (s) by using Cramer’s rule gives F1 (s) F2 (s)

1

−K12 (s) 1 − K22 (s) , U (s) = 1 − K11 (s) −K (s) 12 −K21 (s) 1 − K22 (s) and

− K11 (s) F1 (s) −K21 (s) F2 (s) . V (s) = 1 − K11 (s) −K12 (s) −K21 (s) 1 − K22 (s)

(12.44)

(12.45)

Having determined U (s) and V (s), the unique solution for u(x) and v(x) can be determined by using the inverse Laplace transform method. It is interesting to point out that the computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations of this weakly singular equations. The linear system (12.41) has a unique solution for u(x) and v(x) if and only if 1 − K11 (s) −K12 (s) (12.46) −K21 (s) 1 − K22 (s) = 0. Example 12.5 Solve the system of weakly singular Volterra integral equations by using the Laplace transform method x 3 64 7 2 x 2 (4x + 3) + u(x) = x4 − u(t) + v(t) dt, 1 1 105 (x − t) 2 (x − t) 2 0 x 2 32 7 3 3 2 x (16x − 27) + dt. v(x) = x − 1 u(t) − 1 v(t) 315 (x − t) 2 (x − t) 2 0 (12.47) Taking Laplace transform of both sides of each equation in (12.47) gives * * √ 12 π 6 π π 4! U (s) − 2 V (s) = 5 − +1 , 1−3 9 s s s s s2 (12.48) * * √ 3! 6 π 8 π π U (s) + 1 + 3 V (s) = 4 − 9 −3 . −2 s s s s s2 Solving this system of equations for U (s) and V (s), by using any computer symbolic system, such as Maple or Mathematica, gives

376

12 Systems of Singular Integral Equations

4! 3! , V (s) = 4 . (12.49) s5 s By taking the inverse Laplace transform of both sides of each equation in (12.49), the exact solutions are given by (12.50) (u(x), v(x)) = (x4 , x3 ). U (s) =

Example 12.6 Solve the system of weakly singular Volterra integral equations by using the Laplace transform method 9 2 u(x) = 1 + 2x − x 3 (9x2 + 6x + 10) 20 x 1 2 + dt, 1 u(t) + 1 v(t) (x − t) 3 (x − t) 3 0 (12.51) 3 2 v(x) = 1 + 3x2 + x 3 (27x2 − 72x − 20) 40 x 2 1 dt. + 1 u(t) − 1 v(t) (x − t) 3 (x − t) 3 0 Taking Laplace transforms of both sides of each equation in (12.51) gives

2Γ 23 Γ 23 3Γ 23 1 3 4 1 U (s) − 1 + , + + 1− V (s) = − 2 2 5 s s2 s s2 s3 s3 s3

2Γ 23 Γ 23 Γ 23 1 6 6 6 V (s) = − 1 + . + − U (s) + 1 + − 2 2 5 s s3 s s2 s3 s3 s3 (12.52) Solving this system of equations for U (s) and V (s) we ﬁnd 3 6 1 1 U (s) = + 2 , V (s) = + 3 . (12.53) s s s s The inverse Laplace transform of (12.53) gives the exact solutions by (12.54) (u(x), v(x)) = (1 + 3x, 1 + 3x2 ). Example 12.7 Solve the system of weakly singular Volterra integral equations by using the Laplace transform method 25 8 6 1 u(x) = x + x2 − x 5 (130x 5 + 182x 5 − 210x + 273) 6552 x 1 1 dt, + 1 u(t) + 2 v(t) (x − t) 5 (x − t) 5 0 (12.55) 25 6 6 1 2 5 5 5 v(x) = x − x − x (55x + 66x − 140x + 154) 924 x 1 1 u(t) + v(t) dt. + 3 4 (x − t) 5 (x − t) 5 0

12.3 Systems of the Weakly Singular Volterra Integral Equations

377

Taking Laplace transforms of both sides of each equation in (12.55) and solving the resulting system of equations for U (s) and V (s) we obtain 1 2 1 2 U (s) = 2 + 3 , V (s) = 2 − 3 . (12.56) s s s s The inverse Laplace transform of (12.56) gives the exact solutions by (12.57) (u(x), v(x)) = (x + x2 , x − x2 ). Example 12.8 Use the Laplace transform method to solve the system 3 2 8 3 x 4 (32x2 − 44x + 77) u(x) = 1 + x − x2 + x 3 (9x2 − 12x − 20) − 40 231 x 2 1 dt, + 1 u(t) + 1 v(t) (x − t) 3 (x − t) 4 0 5 4 v(x) = 1 − x + x2 + x 5 (25x2 − 35x − 63) (12.58) 126 6 5 x 6 (72x2 − 102x + 187) + 935 x 2 1 dt. + 1 u(t) − 1 v(t) (x − t) 5 (x − t) 6 0 Taking Laplace transform of both sides of each equation in (12.58), proceeding as before, and solving the resulting system of equations for U (s) and V (s) we obtain 2 1 2 1 1 1 (12.59) U (s) = + 2 − 3 , V (s) = − 2 + 3 . s s s s s s The inverse Laplace transform of (12.59) gives the exact solutions by (u(x), v(x)) = (1 + x − x2 , 1 − x + x2 ).

(12.60)

Exercises 12.3.1 Solve the following systems of weakly singular Volterra integral equations by using the Laplace transform method ⎧ x 1 25 7 1 ⎪ 2 ⎪ 5 ⎪ u(x) = x + x − x + dt 3 u(t) + 3 v(t) ⎪ ⎨ 7 0 (x − t) 5 (x − t) 5 1. x ⎪ ⎪ 1 25 8 1 ⎪ ⎪ dt x5 + ⎩ v(x) = x − x2 − 2 u(t) + 2 v(t) 12 0 (x − t) 5 (x − t) 5 ⎧ x 1 4 3 1 ⎪ 2 ⎪ ⎪ u(x) = x − dt x 2 (4x + 5) + 1 u(t) + 1 v(t) ⎪ ⎨ 15 0 (x − t) 2 (x − t) 2 2. x ⎪ ⎪ 4 3 1 1 ⎪ ⎪ dt x 2 (4x − 5) + ⎩ v(x) = x − 1 u(t) − 1 v(t) 15 0 (x − t) 2 (x − t) 2

378 ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨ 3.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) = ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨

4.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) = ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨

5.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) = ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨

6.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) = ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨

7.

8.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) =

12 Systems of Singular Integral Equations x 1 5 4 1 2 1 + x − x5 + dt 1 u(t) + 1 v(t) 2 0 (x − t) 5 (x − t) 5 x 5 4 1 1 u(t) + v(t) dt 1 − x2 − x 5 + 1 1 2 0 (x − t) 5 (x − t) 5 √ x 2 x 1 2 (16x + 10x + 15) + dt 1 + 3x − 1 (u(t) + 2v(t)) 5 0 (x − t) 2 √ x 1 2 x (2u(t) − v(t)) dt 1 + 3x2 + (8x2 − 20x − 5) + 1 5 0 (x − t) 2 x √ 1 (u(t) + v(t)) dt 1 + x + x2 − 4 x + 1 0 (x − t) 2 x 1 8 3 x 2 (4x + 5) + 1 − x − x2 − (u(t) − v(t)) dt 1 15 0 (x − t) 2 x √ 1 (u(t) + 2v(t)) dt 1 + x + 2x2 + 2 x(2x − 3) + 1 0 (x − t) 2 √ x 2 x 1 2 2 dt (8x + 8x + 3) + 1 − 2x − x − 1 (2u(t) − v(t)) 3 0 (x − t) 2 x 9 2 1 (u(t) + 2v(t)) dt 1 + x − x2 − x 3 (3x2 − 4x + 20) + 1 40 0 (x − t) 3 x 1 3 2 2 2 3 x (27x − 36x − 20) + 1−x+x + dt 1 (2u(t) − v(t)) 40 0 (x − t) 3

⎧ 7 1 9 1 ⎪ x 3 (21x 3 + 280x 3 − 60x2 + 560) ⎪ u(x) = 6 + x2 − ⎪ ⎪ 280 ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ dt + 1 u(t) + 2 v(t) ⎪ ⎨ 0 (x − t) 3 (x − t) 3

11 1 5 1 ⎪ ⎪ ⎪ v(x) = 6 − x2 − x 5 (275x 5 + 2772x 5 − 700x2 + 5544) ⎪ ⎪ 924 ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ dt + ⎩ 3 u(t) + 4 v(t) 0 (x − t) 5 (x − t) 5

12.3.2 The Adomian Decomposition Method The Adomian decomposition method, as presented before, decomposes each solution as an inﬁnite sum of components, where these components are determined recurrently. This method can be used in its standard form, or combined with the noise terms phenomenon. It will be shown that the modiﬁed decomposition method is eﬀective and reliable in handling the systems of weakly singular Volterra integral equations. In view of this, the modiﬁed decomposition method will be used extensively in this section.

12.3 Systems of the Weakly Singular Volterra Integral Equations

379

We will focus our work on the system of the generalized weakly singular Volterra integral equations in two unknowns of the form x u(x) = f1 (x) + (K11 (x, t)u(t) + K12 (x, t)v(t)) dt, 0 (12.61) x (K21 (x, t)u(t) + K22 (x, t)v(t)) dt. v(x) = f2 (x) + 0

The kernels Kij (x, t), 1 i, j 2 and the functions fi (x), i = 1, 2 are given real-valued functions. The kernels Kij are singular kernels of the generalized form given by 1 Kij = 1 i, j 2. (12.62) α , [g(x) − g(t)] ij This type of systems arise in many mathematical physics and chemistry applications such as stereology, heat conduction, crystal growth and the radiation of heat from a semi-inﬁnite solid. The Adomian decomposition method [8] has been discussed extensively in this text. As will be seen later, the modiﬁed decomposition method is reliable and eﬀective in handling the system (12.61). In addition, the noise terms phenomenon will be used when noise terms appear. For revision purposes, we give a brief review of the modiﬁed decomposition method. In this method we usually split each of the source terms f1 (x) and f2 (x) into two parts fi1 (x) deﬁned by f1 (x) = f11 + f12 , f2 (x) = f21 + f22 . (12.63) Based on this, we use the modiﬁed recurrence relation as follows: u0 (x) = f11 (x),

v0 (x) = f21 (x), x (K11 (x, t)u0 (t) + K12 (x, t)v0 (t)) dt, u1 (x) = f12 (x) + v1 (x) = f22 (x) +

uk+1 (x) = vk+1 (x) =

0

x 0

(K21 (x, t)u0 (t) + K22 (x, t)v0 (t)) dt,

(12.64)

x

0

0

(K11 (x, t)uk (t) + K12 (x, t)vk (t)) dt,

k 1,

(K21 (x, t)uk (t) + K22 (x, t)vk (t)) dt,

k 1.

x

The following examples will illustrate the analysis presented above. Example 12.9 Use the modiﬁed method to solve the system of the generalized weakly singular integral equations

380

12 Systems of Singular Integral Equations

x 3 1 1 2 x x u(x) = e − (3e + 7)(e − 1) 3 + dt, 1 u(t) + 1 v(t) 10 (ex − et ) 3 (ex − et ) 3 0 x 1 9 1 5 v(x) = e2x + (ex − 1) 3 + u(t) − v(t) dt. 1 1 10 (ex − et ) 3 (ex − et ) 3 0 (12.65) We set the modiﬁed recurrence relation u0 (x) = ex , v0 (x) = e2x , x 3 1 1 2 x x 3 dt u1 (x) = − (3e + 7)(e − 1) + 1 u0 (t) + 1 v0 (t) 10 (ex − et ) 3 (ex − et ) 3 0 x

= 0, x 9 x 1 1 5 3 (e − 1) + v1 (x) = dt = 0. 1 u0 (t) − 1 v0 (t) 10 (ex − et ) 3 (ex − et ) 3 0 (12.66) The exact solutions are therefore given by (u(x), v(x)) = (ex , e2x ).

(12.67)

Example 12.10 Solve the system of the generalized weakly singular Volterra integral equations by using the modiﬁed decomposition method 2 3 3 2 u(x) = sin x + (cos x − 1) 3 − sin 3 x 2 2 x 1 1 + u(t) + v(t) dt, 1 1 (cos x − cos t) 3 (sin x − sin t) 3 0 (12.68) 1

1

v(x) = cos x + 3(cos x − 1) 3 − 3 sin 3 x x 1 1 + dt. 2 u(t) − 2 v(t) (cos x − cos t) 3 (sin x − sin t) 3 0 Proceeding as before, we set the modiﬁed recurrence relation u0 (x) = sin x, v0 (x) = cos x, 2 3 3 2 u1 (x) = (cos x − 1) 3 − sin 3 x 2 2 x 1 1 u (t) + v (t) dt = 0, + 0 0 1 1 (cos x − cos t) 3 (sin x − sin t) 3 0 1

1

v1 (x) = 3(cos x − 1) 3 − 3 sin 3 x x 1 1 dt = 0, + 2 u0 (t) − 2 v0 (t) (cos x − cos t) 3 (sin x − sin t) 3 0 uk+1 (x) = 0,

vk+1 (x) = 0,

k 1. (12.69)

The exact solutions are therefore given by (u(x), v(x)) = (sin x, cos x).

(12.70)

12.3 Systems of the Weakly Singular Volterra Integral Equations

381

Example 12.11 Solve the system of the generalized weakly singular Volterra integral equations by using the modiﬁed decomposition method x 5 4 5 u(x) = x4 − 2x 2 − x4 + u(t) + v(t) dt, 1 1 (x5 − t5 ) 2 (x6 − t6 ) 3 0 x 15 24 15 24 5 4 5 v(x) = x − 4x + 5x + dt. 1 u(t) − 1 v(t) (x5 − t5 ) 4 (x6 − t6 ) 5 0 (12.71) We next set the modiﬁed recurrence relation v0 (x) = x5 , u0 (x) = x4 , u1 (x) = 0, v1 (x) = 0, (12.72) uk+1 (x) = 0, vk+1 (x) = 0, k 1. The exact solutions are therefore given by (u(x), v(x)) = (x4 , x5 ).

(12.73)

Example 12.12 Solve the system of the generalized weakly singular Volterra integral equations by using the modiﬁed decomposition method 1

1

u(x) = ex − 2(ex − 1) 2 (1 + e− 2 x ) x 1 1 + dt, 1 u(t) + 1 v(t) (ex − et ) 2 (e−t − e−x ) 2 0 3 2 2 v(x) = e−x − (ex − 1) 3 (1 + e− 3 x ) 2 x 1 1 dt. + 1 u(t) + 1 v(t) (ex − et ) 3 (e−t − e−x ) 3 0 Now we use the modiﬁed recurrence relation u0 (x) = ex , v0 (x) = e−x , u1 (x) = 0, v1 (x) = 0, uk+1 (x) = 0, vk+1 (x) = 0, k 1. The exact solutions are therefore given by (u(x), v(x)) = (ex , e−x ).

(12.74)

(12.75)

(12.76)

Exercises 12.3.2 Solve the following systems of generalized weakly singular integral equations by using the modiﬁed decomposition method

382 ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨ 1.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) = ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨

2.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) = ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨

3.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) =

12 Systems of Singular Integral Equations x 1 π 2 2 2 1 + x − (x − 6) + dt 1 u(t) + 1 v(t) 4 0 (x2 − t2 ) 2 (x2 − t2 ) 2 x 9 2 1 4 u(t) − v(t) dt 1 − x2 − x 3 (3x2 + 4) + 1 1 8 0 (x − t) 3 (x − t) 3 x 1 2 1 dt x − 4x 2 (x + 1) + 1 u(t) + 1 v(t) 0 (x − t) 2 (x − t) 2 x 4 1 3 2 u(t) − v(t) dt 1+x− x 3 (9x − 5) + 1 1 10 0 (x − t) 3 (x − t) 3 x 1 4 4 3 u(t) + v(t) dt x− x 2 (16x + 5) + 1 1 15 0 (x − t) 2 (x − t) 2 x 1 9 5 1 x 3 (3x − 4) + x4 + u(t) − v(t) dt 1 1 40 0 (x − t) 3 (x − t) 3

⎧ 1 1 ⎪ u(x) = sin x + 2(cos x − 1) 2 − 2 sin 2 x ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ + dt, ⎪ 1 u(t) + 1 v(t) ⎪ ⎪ 0 (cos x − cos t) 2 (sin x − sin t) 2 ⎨ 4. 2 2 3 3 ⎪ v(x) = cos x + (cos x − 1) 3 + sin 3 x ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ + dt. ⎪ 1 u(t) − 1 v(t) ⎩ 0 (cos x − cos t) 3 (sin x − sin t) 3 ⎧ 2 1 3 ⎪ ⎪ u(x) = cos x + (cos x − 1) 3 − 2 sin 2 x ⎪ ⎪ 2 ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ + u(t) + v(t) dt, ⎪ 1 1 ⎪ ⎨ 0 (sin x − sin t) 2 (cos x − cos t) 3 5. 4 3 5 4 ⎪ ⎪ v(x) = sin x + (cos x − 1) 5 − sin 4 x ⎪ ⎪ ⎪ 4 3 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ dt. + 1 u(t) + 1 v(t) ⎩ 0 (sin x − sin t) 4 (cos x − cos t) 5 ⎧ 2 2 3 3 ⎪ ⎪ u(x) = sinh x − (cosh x − 1) 3 − sinh 3 x ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ + dt, ⎨ 1 u(t) + 1 v(t) 0 (cosh x − cosh t) 3 (sinh x − sinh t) 3 6. ⎪ 1 1 ⎪ ⎪ v(x) = cosh x − 3(cosh x − 1) 3 − 3 sinh 3 x ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ + dt. ⎪ 2 u(t) + 2 v(t) ⎩ 0 (cosh x − cosh t) 3 (sinh x − sinh t) 3

References

7.

383

⎧ 2 1 3 ⎪ u(x) = cosh x − (cosh x − 1) 3 − 2 sinh 2 x ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ + dt, 1 u(t) + 1 v(t) ⎪ ⎨ 0 (sinh x − sinh t) 2 (cosh x − cosh t) 3

4 3 4 5 ⎪ ⎪ ⎪ v(x) = sinh x − (cosh x − 1) 5 − sinh 4 x ⎪ ⎪ 4 3 ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ + dt. ⎩ 1 u(t) + 1 v(t) 0 (sinh x − sinh t) 4 (cosh x − cosh t) 5 ⎧ 2 3 −2x ⎪ ⎪ u(x) = ex − (ex − 1) 3 (1 + e 3 ) ⎪ ⎪ 2 ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ u(t) + v(t) dt, + 1 1 ⎪ ⎨ 0 (ex − et ) 3 (e−t − e−x ) 3 8. 3 3 4 ⎪ ⎪ v(x) = e−x − (ex − 1) 4 (1 − e− 4 x ) ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ + u(t) − v(t) dt ⎩ 1 1 0 (ex − et ) 4 (e−t − e−x ) 4

References 1. E. Erdogan, Approximate solutions of singular integral equations, SIAM J. Appl. Math., 17 (1969) 1041–1059 2. R. Estrada and R. Kanwal, Singular Integral Equations, Birkhauser, Berlin, (2000). 3. A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 4. R.P. Kanwal, Linear Integral Equations, Birkhauser, Boston, (1997). 5. E.G. Ladopoulos, Singular Integra Equations, Springer, Berlin, (2000). 6. M. Masujima, Applied Mathematical Methods in Theoretical Physics, WileyVCH, Weinheim, (2005). 7. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).

Part II

Nonlinear Integral Equations

Chapter 13

Nonlinear Volterra Integral Equations

13.1 Introduction It is well known that linear and nonlinear Volterra integral equations arise in many scientiﬁc ﬁelds such as the population dynamics, spread of epidemics, and semi-conductor devices. Volterra started working on integral equations in 1884, but his serious study began in 1896. The name integral equation was given by du Bois-Reymond in 1888. However, the name Volterra integral equation was ﬁrst coined by Lalesco in 1908. The linear Volterra integral equations and the linear Volterra integrodiﬀerential equations were presented in Chapters 3 and 5 respectively. It is our goal in this chapter to study the nonlinear Volterra integral equations of the ﬁrst and the second kind. The nonlinear Volterra equations are characterized by at least one variable limit of integration. In the nonlinear Volterra integral equations of the second kind, the unknown function u(x) appears inside and outside the integral sign. The nonlinear Volterra integral equation of the second kind is represented by the form x K(x, t)F (u(t))dt. (13.1) u(x) = f (x) + 0

However, the nonlinear Volterra integral equations of the ﬁrst kind contains the nonlinear function F (u(x)) inside the integral sign. The nonlinear Volterra integral equation of the ﬁrst kind is expressed in the form x f (x) = K(x, t)F (u(t))dt. (13.2) 0

For these two kinds of equations, the kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x) such as u2 (x), sin(u(x)), and eu(x) .

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

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13 Nonlinear Volterra Integral Equations

13.2 Existence of the Solution for Nonlinear Volterra Integral Equations In this section we will present an existence theorem for the solution of nonlinear Volterra integral equations. The complete proof of this theorem can be found in [1–6]. However, in what follows, we present a brief summary of the conditions under which a solution exists for this equation. We ﬁrst rewrite the nonlinear Volterra integral equation of the second kind by x

u(x) = f (x) +

G(x, t, u(t))dt.

(13.3)

0

The speciﬁc conditions under which a solution exists for the nonlinear Volterra integral equation are: (i) The function f (x) is integrable and bounded in a x b. (ii) The function f (x) must satisfy the Lipschitz condition in the interval (a, b). This means that |f (x) − f (y)| < k|x − y|. (13.4) (iii) The function G(x, t, u(t)) is integrable and bounded |G(x, t, u(t))| < K in a x, t b. (iv) The function G(x, t, u(t)) must satisfy the Lipschitz condition (13.5) |G(x, t, z) − G(x, t, z )| < M |z − z |. The emphasis in this chapter will be on solving the nonlinear Volterra integral equations rather than proving theoretical concepts of convergence and existence. The theorems of uniqueness, existence, and convergence are important and can be found in the literature. The concern in this text will be on the determination of the solution u(x) of the nonlinear Volterra integral equation of the ﬁrst and the second kind.

13.3 Nonlinear Volterra Integral Equations of the Second Kind We begin our study on nonlinear Volterra integral equations of the second kindgiven by x K(x, t)F (u(t))dt, (13.6) u(x) = f (x) + 0

where the kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x) such as u3 (x), cos(u(x)), andeu(x) . The unknown function u(x), that will be determined, occurs inside and outside the integral sign.

13.3 Nonlinear Volterra Integral Equations of the Second Kind

389

The nonlinear Volterra equation (13.6) will be handled by using three distinct methods. The three methods are the successive approximations method, the series solution method, and the Adomian decomposition method (ADM). The latter will be combined with the modiﬁed decomposition method and the noise terms phenomenon.

13.3.1 The Successive Approximations Method The successive approximations method [7], or the Picard iteration method was used before in Chapters 3 and 4. This method solves any problem by ﬁnding successive approximations to the solution by starting with an initial guess, called the zeroth approximation. As will be seen later, the zeroth approximation is any selective real-valued function that will be used in a recurrence relation to determine the other approximations. Given the nonlinear Volterra integral equation of the second kind x u(x) = f (x) + K(x, t)F (u(t))dt, (13.7) 0

where u(x) is the unknown function to be determined and K(x, t) is the kernel. The successive approximations method introduces the recurrence relation x K(x, t)F (un (t))dt, n 0, (13.8) un+1 (x) = f (x) + 0

where the zeroth approximation u0 (x) can be any selective real valued function. We always start with an initial guess for u0 (x), mostly we select 0, 1, or x for u0 (x). Using this selection of u0 (x) into (13.8), several successive approximations uk , k 1 will be determined as x u1 (x) = f (x) + K(x, t)F (u0 (t))dt, 0

u2 (x) = f (x) +

.. .

x

K(x, t)F (u2 (t))dt,

0

un+1 (x) = f (x) +

K(x, t)F (u1 (t))dt,

0

u3 (x) = f (x) +

x

(13.9)

x

0

K(x, t)F (un (t))dt.

Consequently, the solution u(x) is obtained by using u(x) = lim un+1 (x). n→∞

(13.10)

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13 Nonlinear Volterra Integral Equations

The question of convergence of un+1 (x) was examined in Chapter 3. The successive approximations method, or the Picard iteration method will be illustrated by the following examples. Example 13.1 Use the successive approximations method to solve the nonlinear Volterra integral equation x 1 x 3x u(x) = e + x(1 − e ) + xu3 (t)dt. (13.11) 3 0 For the zeroth approximation u0 (x), we can select (13.12) u0 (x) = 1. The method of successive approximations admits the use of the iteration formula x 1 x 3x xu3n (t)dt, n 0. (13.13) un+1 (x) = e + x(1 − e ) + 3 0 Substituting (13.12) into (13.13) we obtain the approximations u0 (x) = 1, x 1 x 3x u1 (x) = e + x(1 − e ) + xu30 (t)dt 3 0 4 35 67 1 = 1 + x + x2 − x3 − x4 − x5 + · · · , 2! 3 24 60 x 1 u2 (x) = ex + x(1 − e3x ) + xu31 (t)dt (13.14) 3 0 1 1 1 67 = 1 + x + x2 + x3 + x4 − x5 + · · · , 2! 3! 4! 60 x 1 u3 (x) = ex + x(1 − e3x ) + xu32 (t)dt 3 0 1 1 1 1 1 = 1 + x + x2 + x3 + x4 + x5 + x6 + · · · , 2! 3! 4! 5! 6! and so on. Consequently, the solution u(x) of (13.11) is given by u(x) = lim un (x) = ex . n→∞

(13.15)

Example 13.2 Use the successive approximations method to solve the nonlinear Volterra integral equation x 16 4 u(x) = 4x − x3 − x4 + (x − t + 1)u2 (t)dt. (13.16) 3 3 0 For the zeroth approximation u0 (x), we can select u0 (x) = 0.

(13.17)

13.3 Nonlinear Volterra Integral Equations of the Second Kind

391

The method of successive approximations admits the use of the iteration formula x 16 4 un+1 (x) = 4x − x3 − x4 + (x − t + 1)u2n (t)dt, n 0. (13.18) 3 3 0 Substituting (13.17) into (13.18) we obtain the approximations u0 (x) = 0, x 16 3 4 4 u1 (x) = 4x − x − x + (x − t + 1)u20 (t)dt 3 3 0 4 16 = 4x − x3 − x4 , 3 3 x 16 3 4 4 u2 (x) = 4x − x − x + (x − t + 1)u21 (t)dt 3 3 0 4 4 4 4 128 5 16 6 16 3 16 3 x − x + x − x − x − x + ··· , = 4x + 3 3 3 3 15 5 x 16 3 4 4 (x − t + 1)u22 (t)dt u3 (x) = 4x − x − x + 3 3 0 128 5 128 5 16 6 16 6 x − x + x − x + ··· . = 4x + 15 15 5 5 (13.19) By canceling the noise terms, the solution u(x) of (13.16) is given by (13.20) u(x) = lim un (x) = 4x. n→∞

Example 13.3 Use the successive approximations method to solve the nonlinear Volterra integral equation x 1 1 1 (x − t)u2 (t)dt. (13.21) u(x) = cos(x) + cos(2x) − x2 − + 8 4 8 0 For the zeroth approximation u0 (x), we can select (13.22) u0 (x) = 1. The method of successive approximations admits the use of the iteration formula x 1 1 1 (x − t)u2n (t)dt, n 0. (13.23) un+1 (x) = cos(x) + cos(2x) − x2 − + 8 4 8 0 Substituting (13.22) into (13.23) we obtain the approximations u0 (x) = 1, 1 1 1 u1 (x) = 1 − x2 + x4 − x6 + · · · , 2! 8 80 (13.24) 1 2 1 4 1 6 u2 (x) = 1 − x + x + x + ··· , 2! 4! 240 1 2 1 4 1 u3 (x) = 1 − x + x − x6 + · · · . 2! 4! 6!

392

13 Nonlinear Volterra Integral Equations

Consequently, the solution u(x) of (13.21) is given by u(x) = lim un (x) = cos x. n→∞

(13.25)

Example 13.4 Use the successive approximations method to solve the nonlinear Volterra integral equation x 1 2 1 4 1 6 u(x) = 1 + x − x − x + (x − t)u2 (t)dt. (13.26) 2 6 30 0 For the zeroth approximation u0 (x), we can select u0 (x) = 1.

(13.27)

The method of successive approximations admits the use of the iteration formula x 1 1 1 (x − t)u2n (t)dt, n 0. (13.28) un+1 (x) = 1 + x2 − x4 − x6 + 2 6 30 0 Substituting (13.27) into (13.28) we obtain the approximations u0 (x) = 1, 1 1 u1 (x) = 1 + x2 − x4 − x6 , 6 30 1 4 1 4 1 (13.29) u2 (x) = 1 + x2 + x − x − x6 + · · · , 6 6 90 1 4 1 4 1 6 1 6 2 u3 (x) = 1 + x + x − x + x − x + ··· , 6 6 90 90 and so on. By canceling the noise terms, the solution u(x) of (13.26) is given by (13.30) u(x) = lim un (x) = 1 + x2 . n→∞

Exercises 13.3.1 Solve the following nonlinear Volterra integral equations by using the successive approximations method x 1. u(x) = 1 + u2 (t)dt, |x| < 1 0

2. u(x) = 1 + 3x −

3 1 2 x − x3 − x4 + 2 4

3. u(x) = 1 + 2x −

7 2 3 x − 4x3 − x4 + 2 4

x 0

(x − t)u2 (t)dt

x

0 x

(x − t + 1)u2 (t)dt

1 3 1 5 (x − t − 1)u3 (t)dt x − x + 2 20 0 x 1 1 (x − t)u2 (t)dt 5. u(x) = sin x + sin2 x − x2 + 4 4 0 4. u(x) = 1 + 2x + x2 +

13.3 Nonlinear Volterra Integral Equations of the Second Kind x 1 1 1 6. u(x) = sin x + cos x + sin 2x − x − x2 + (x − t)u2 (t)dt 4 2 2 0 x 1 1 1 7. u(x) = cos x − sin x − sin 2x + x − x2 + (x − t)u2 (t)dt 4 2 2 0 x 3 1 5 8. u(x) = 3 cos x + cos2 x − − x2 + (x − t)u2 (t)dt 4 4 4 0 x 5 1 1 (x − t)u2 (t)dt 9. u(x) = sinh x − cosh2 x + − 2x − x2 + 4 4 4 0 x 1 1 1 10. u(x) = ex − e2x + + x + (x − t)u2 (t)dt 4 4 2 0 x 1 1 1 11. u(x) = ex − e3x + + x + (x − t)u3 (t)dt 9 9 3 0 x 7 1 1 3 12. u(x) = ex + e2x + + x − x2 + (x − t − 1)u2 (t)dt 4 4 2 2 0

393

13.3.2 The Series Solution Method The series solution method was applied in Chapters 3, 4 and 5 to handle linear Volterra and Fredholm integral equations [1–2]. In this section, the series solution method will be applied in a similar manner to handle the nonlinear Volterra integral equations. Recall that the generic form of Taylor series at x = 0 can be written as ∞ u(x) = a n xn . (13.31) n=0

We will assume that the solution u(x) of the nonlinear Volterra integral equation x K(x, t)F (u(t))dt,

u(x) = f (x) +

(13.32)

0

is analytic, and therefore possesses a Taylor series of the form given in (13.31), where the coeﬃcients an will be determined recurrently. Substituting (13.31) into both sides of (13.32) gives ∞ x ∞ n n an x = T (f (x)) + K(x, t) F an t dt, (13.33) 0

n=0

or for simplicity we use 2

a0 + a1 x + a2 x + · · · = T (f (x)) +

n=0

0

x

K(x, t) F (a0 + a1 t + a2 t2 + · · · ) dt,

(13.34) where T (f (x)) is the Taylor series for f (x). The integral equation (13.32) will be converted to a traditional integral in (13.33) or (13.34) where instead of integrating the nonlinear term F (u(x)), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x)

394

13 Nonlinear Volterra Integral Equations

includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integral in (13.33) or (13.34), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x in both sides of the resulting equation to obtain a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (13.31). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we determine, the higher accuracy level we achieve. Example 13.5 Solve the following nonlinear Volterra integral equation by using the series solution method x 1 1 1 (x − t)u2 (t)dt. (13.35) u(x) = 1 + x − x2 − x3 − x4 + 2 3 12 0 Using the series form (13.31) into both sides of (13.35) gives 1 1 1 a0 + a1 x + a2 x2 + · · · = 1 + x − x2 − x3 − x4 2 3 12 x

2 + (x − t) a0 + a1 t + a2 t2 + a3 t3 + · · · dt, 0

(13.36) where by integrating the integral at the right side, and collecting like powers of x we obtain a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · 1 1 1 = 1 + x + (a20 − 1)x2 + (a0 a1 − 1)x3 + (a21 + 2a0 a2 − 1)x4 + · · · . 2 3 12 (13.37) Equating the coeﬃcients of like powers of x in both sides yields a0 = 1, a1 = 1, The exact solution is given by

an = 0,

u(x) = 1 + x.

for n 2.

(13.38) (13.39)

Example 13.6 We next consider the nonlinear Volterra integral equation x 1 3x 1 1 x (x − t)u3 (t)dt. (13.40) u(x) = + x + e − e + 9 3 9 0 Substituting the series (13.31) into both sides of (13.40) noting that u3 (x) = a0 3 +3a0 2 a1 x+(3a0 a1 2 +3a0 2 a2 )x2 +(a31 +6a1 a2 a0 +3a3 a20 )x3 +· · · , (13.41)

13.3 Nonlinear Volterra Integral Equations of the Second Kind

395

gives 1 1 1 a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · = + x + ex − e3x 9 3 9 x

3 + (x − t) a0 + a1 t + a2 t2 + a3 t3 + · · · dt.

(13.42)

0

Integrating the integral at the right hand side of (13.42), using the Taylor series of ex and e3x , and equating the coeﬃcients of like powers of x we ﬁnd a0 = 1, a1 = 1, 1 1 a2 = a20 = , 2! 2! 1 1 1 a3 = − + a20 a1 = , 3 2 3! (13.43) 1 1 1 2 a4 = − + a0 (a1 + a0 a2 ) = , 3 4 4! .. . 1 n! This gives the solution in a series form 1 1 1 u(x) = 1 + x + x2 + x3 + x4 + · · · . 2! 3! 4! Consequently, the exact solution is given by an =

u(x) = ex .

(13.44) (13.45)

Example 13.7 We next consider the nonlinear Volterra integral equation x 1 1 2 (x − t)u2 (t)dt. (13.46) u(x) = (x − x ) + cos x − sin x − sin(2x) + 2 4 0 Substituting the series (13.31) into both sides of (13.46) gives 1 a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · = (x − x2 ) + cos x − sin x 2 x (13.47)

2 1 (x − t) a0 + a1 t + a2 t2 + a3 t3 + · · · dt. − sin 2x + 4 0 Integrating the integral at the right hand side of (13.47), using the Taylor series of sin x, cos x and sin 2x, and equating the coeﬃcients of like powers of x we ﬁnd a0 = 1, a1 = −1, 1 1 a2 = −1 + a20 = − , 2! 2! 1 1 1 a 3 = + a0 a 1 = , (13.48) 2 3 3!

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13 Nonlinear Volterra Integral Equations

1 1 1 1 + a0 a2 + a21 = , 4! 6 12 4! 1 3 1 1 a5 = − + a1 a2 + a0 a3 = − , 40 10 10 5! and so on. The solution in a series form is given by 1 1 1 1 u(x) = 1 − x2 + x4 + · · · − x − x3 + x5 + · · · , 2! 4! 3! 5! that converges to the exact solution u(x) = cos x − sin x. a4 =

(13.49)

(13.50)

Example 13.8 We ﬁnally consider the nonlinear Volterra integral equation x 1 9 1 2 (x − t)u2 (t)dt. (13.51) u(x) = + 2x − x − sinh x − cosh 2x + 8 4 8 0 Substituting the series (13.31) into both sides of (13.51) gives 9 1 a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · = + 2x − x2 − sinh x 8 4 x (13.52)

2 1 − cosh 2x + (x − t) a0 + a1 t + a2 t2 + a3 t3 + · · · dt. 8 0 Integrating the integral at the right hand side of (13.52), using the Taylor series of sinh x and cosh 2x, and equating the coeﬃcients of like powers of x we ﬁnd a0 = 1, a1 = 1, 1 1 a2 = − + a20 = 0, 2 2! 1 1 1 a 3 = − + a0 a 1 = , (13.53) 6 3 3! 1 1 1 a4 = − + a0 a2 + a21 = 0, 12 6 12 1 1 1 1 a5 = − + a1 a2 + a0 a3 = , 120 10 10 5! and so on. The solution in a series form is given by 1 5 1 3 (13.54) u(x) = 1 + x + x + x + · · · , 3! 5! that converges to the exact solution u(x) = 1 + sinh x. (13.55)

Exercises 13.3.2 Solve the following nonlinear Volterra integral equations by using the series solution method

13.3 Nonlinear Volterra Integral Equations of the Second Kind x 1 1 1 4 1. u(x) = 1 + x − x2 − x3 − (x − t)u2 (t)dt x + 2 3 12 0 x 1 1 1 6 x + 2. u(x) = 1 + x2 − x4 − (x − t)u2 (t)dt 2 6 30 0 x 7 3 3. u(x) = 1 + 2x − x2 − 4x3 − x4 + (x − t + 1)u2 (t)dt 2 4 0 x 2 1 (x − t)u3 (t)dt 4. u(x) = 1 + 2x − x2 − x3 − x4 − x5 + 2 5 0 x 1 1 5. u(x) = sin x + sin2 x − x2 + (x − t)u2 (t)dt 4 4 0 x 1 1 1 6. u(x) = sin x + cos x + sin 2x − x − x2 + (x − t)u2 (t)dt 4 2 2 0 x 1 1 1 7. u(x) = − x2 + cosh x − cosh 2x + (x − t)u2 (t)dt 8 4 8 0 x 1 1 1 8. u(x) = x − x3 + cosh x − sinh 2x + (x − t)2 u2 (t)dt 4 6 8 0 x 5 1 1 (x − t)u2 (t)dt 9. u(x) = sinh x − cosh2 x + − 2x − x2 + 4 4 4 0 x 1 1 1 1 10. u(x) = + x + x2 + ex − e2x + (x − t)2 u2 (t)dt 4 2 2 4 0 x 1 1 1 11. u(x) = ex − e3x + + x + (x − t)u3 (t)dt 9 9 3 0 x 1 1 1 12. u(x) = − x + e−x − e−2x + (x − t)u2 (t)dt 4 2 4 0

397

13.3.3 The Adomian Decomposition Method The Adomian decomposition method has been outlined before in previous chapters and has been applied to a wide class of linear Volterra and Fredholm integral equations. The method usually decomposes the unknown function u(x) into an inﬁnite sum of components that will be determined recursively through iterations as discussed before. The Adomian decomposition method will be applied in this chapter and in the coming chapters to handle nonlinear integral equations. Although the linear term u(x) is represented by an inﬁnite sum of components, the nonlinear terms such as u2 , u3 , u4 , sin u, eu , etc. that appear in the equation, should be expressed by a special representation, called the Adomian polynomials An , n 0. Adomian introduced a formal algorithm to establish a reliable representation for all forms of nonlinear terms. Other techniques to evaluate Adomian polynomials were developed, but Adomian technique remains the commonly used one. In this text, we will use the Adomian algorithm to evaluate Adomian polynomials. The representation of the nonlinear

398

13 Nonlinear Volterra Integral Equations

terms by Adomian polynomials is necessary to handle the nonlinear integral equations in a reliable way. In the following, the Adomian algorithm for calculating the so-called Adomian polynomials for representing nonlinear terms will be introduced in details. The algorithm will be explained by illustrative examples that will cover a wide variety of nonlinear forms.

Calculation of Adomian Polynomials The Adomian decomposition method [8] assumes that the unknown linear function u(x) may be represented by the inﬁnite decomposition series ∞ u(x) = un (x), (13.56) n=0

where the components un (x), n 0 will be computed in a recursive way. However, the nonlinear term F (u(x)), such as u2 , u3 , u4 , sin u, eu, etc. can be expressed by an inﬁnite series of the so-called Adomian polynomials An given by n 1 dn i F An = λ ui , n = 0, 1, 2, · · · , (13.57) n! dλn i=0 λ=0

where the so-called Adomian polynomials An can be evaluated for all forms of nonlinearity. The general formula (13.57) can be easily used as follows. Assuming that the nonlinear function is F (u(x)), therefore by using (13.57), Adomian polynomials are given by A0 = F (u0 ), A1 = u1 F (u0 ), 1 A2 = u2 F (u0 ) + u21 F (u0 ), 2! 1 (13.58) A3 = u3 F (u0 ) + u1 u2 F (u0 ) + u31 F (u0 ), 3! 1 2 1 A4 = u4 F (u0 ) + u2 + u1 u3 F (u0 ) + u21 u2 F (u0 ) 2! 2! 1 + u41 F (iv) (u0 ). 4! Two important observations can be made here. First, A0 depends only on u0 , A1 depends only on u0 and u1 , A2 depends only on u0 , u1 , and u2 , and so on. Second, substituting (13.58) into (13.57) gives F (u) = A0 + A1 + A2 + A3 + · · · = F (u0 ) + (u1 + u2 + u3 + · · · )F (u0 ) 1 + (u21 + 2u1 u2 + 2u1 u3 + u22 + · · · )F (u0 ) + · · · 2!

13.3 Nonlinear Volterra Integral Equations of the Second Kind

399

1 3 (u + 3u21 u2 + 3u21 u3 + 6u1 u2 u3 + · · · )F (u0 ) + · · · 3! 1 1 (13.59) = F (u0 ) + (u − u0 )F (u0 ) + (u − u0 )2 F (u0 ) + · · · . 2! The last expansion conﬁrms a fact that the series in An polynomials is a Taylor series about a function u0 and not about a point as in the standard Taylor series. The few Adomian polynomials given above in (13.58) clearly show that the sum of the subscripts of the components of u(x) of each term of An is equal to n. In the following, we will calculate Adomian polynomials for several nonlinear terms that may arise in nonlinear integral equations. +

Case 1. The ﬁrst four Adomian polynomials for F (u) = u2 are given by A0 = F (u0 ) = u20 , A1 = u1 F (u0 ) = 2u0 u1 , 1 A2 = u2 F (u0 ) + u21 F (u0 ) = 2u0 u2 + u21 , 2! 1 A3 = u3 F (u0 ) + u1 u2 F (u0 ) + u31 F (u0 ) = 2u0 u3 + 2u1 u2 . 3! Case 2. The ﬁrst four Adomian polynomials for F (u) = u3 are given by A0 = F (u0 ) = u30 , A1 = u1 F (u0 ) = 3u20 u1 , 1 A2 = u2 F (u0 ) + u21 F (u0 ) = 3u20 u2 + 3u0 u21 , 2! 1 A3 = u3 F (u0 ) + u1 u2 F (u0 ) + u31 F (u0 ) = 3u20 u3 + 6u0 u1 u2 + u31 , 3! 1 1 1 A4 = u4 F (u0 ) + u2 + u1 u3 F (u0 ) + u21 u2 F (u0 ) + u41 F (iv) (u0 ). 2! 2 2! 4! Case 3. The ﬁrst four Adomian polynomials for F (u) = u4 are given by A0 A1 A2 A3

= u40 , = 4u30 u1 , = 4u30 u2 + 6u20 u21 , = 4u30 u3 + 4u31 u0 + 12u20 u1 u2 .

Case 4. The ﬁrst four Adomian polynomials for F (u) = sin u are given by A0 = sin u0 ,

400

13 Nonlinear Volterra Integral Equations

A1 = u1 cos u0 , A2 = u2 cos u0 −

1 2 u sin u0 , 2! 1

A3 = u3 cos u0 − u1 u2 sin u0 −

1 3 u cos u0 . 3! 1

Case 5. The ﬁrst four Adomian polynomials for F (u) = cos u are given by A0 = cos u0 , A1 = −u1 sin u0 , A2 = −u2 sin u0 −

1 2 u cos u0 , 2! 1

A3 = −u3 sin u0 − u1 u2 cos u0 +

1 3 u sin u0 . 3! 1

Case 6. The ﬁrst four Adomian polynomials for F (u) = eu are given by A0 = eu0 , A1 = u1 eu0 , 1 A2 = u2 + u21 eu0 , 2! 1 A3 = u3 + u1 u2 + u31 eu0 . 3!

Applying the Adomian Decomposition Method In what follows we present an outline for using the Adomian decomposition method for solving the nonlinear Volterra integral equation x K(x, t) F (u(t))dt, (13.60) u(x) = f (x) + 0

where F (u(t)) is a nonlinear function of u(x). The nonlinear Volterra integral equation (13.60) contains the linear term u(x) and the nonlinear function F (u(x)). The linear term u(x) of (13.60) can be represented normally by the decomposition series ∞ u(x) = un (x), (13.61) n=0

where the components un (x), n 0 can be easily computed in a recursive manner as discussed before. However, the nonlinear function F (u(x)) of (13.60) should be represented by the so-called Adomian polynomials An by using the algorithm

13.3 Nonlinear Volterra Integral Equations of the Second Kind

An =

1 dn n! dλn

F

n i=0

401

, n = 0, 1, 2, · · · ,

λi ui

(13.62)

λ=0

that was given above in (13.57). The standard Adomian method will be used for the ﬁrst example. However, for the next three examples, we will use the modiﬁed decomposition method and the noise terms phenomenon to minimize the use of Adomian polynomials. Example 13.9 Use the Adomian decomposition method to solve the nonlinear Volterra integral equation x u(x) = x + u2 (t)dt. (13.63) 0

Substituting the series (13.61) and the Adomian polynomials (13.62) into the left side and the right side of (13.63) respectively gives x ∞ ∞ un (x) = x + An (t)dt, (13.64) n=0

0

n=0 2

where An are the Adomian polynomials for u (x) as shown above. Using the Adomian decomposition method we set x u0 (x) = x, uk+1 (x) = Ak (t)dt, k 0. (13.65) 0

This in turn gives u0 (x) = x, x 1 u20 (t)dt = x3 , u1 (x) = 3 0 x 2 5 (2u0 (t)u1 (t))dt = u2 (x) = x , 15 0 x 17 7 u3 (x) = x , (2u0 (t)u2 (t) + u21 (t))dt = 315 0 and so on. The solution in a series form is given by 2 17 7 1 x + ··· , u(x) = x + x3 + x5 + 3 15 315 that converges to the exact solution u(x) = tan x.

(13.66)

(13.67) (13.68)

Example 13.10 We consider the nonlinear Volterra integral equation x 3 1 (x − t)u2 (t)dt. (13.69) u(x) = 1 + 3x − x2 − x3 − x4 + 2 4 0 Substituting the series (13.61) and the Adomian polynomials (13.62) into the left side and the right side of (13.69) respectively gives

402

13 Nonlinear Volterra Integral Equations ∞

1 3 un (x) = 1 + 3x − x2 − x3 − x4 + 2 4 n=0

0

x

(x − t)

∞

An (t)dt,

(13.70)

n=0

where An are the Adomian polynomials for u2 (x) as shown above. Using the modiﬁed decomposition method we set 1 u0 (x) = 1 + 3x − x2 , 2 x 3 u1 (x) = −x3 − x4 + (x − t)A0 (t)dt 4 0 x 3 1 5 3 1 6 (x − t)u20 (t)dt = x2 − x4 − x5 + = −x3 − x4 + x , 4 2 6 20 120 0 x uk+1 (x) = (x − t)Ak (t)dt, k 1. 0

(13.71) We can observe the appearance of the noise terms − 12 x2 and 12 x2 between u0 (x) and u1 (x). By canceling the noise term − 12 x2 from u0 (x), we can show that u(x) = 1 + 3x, (13.72) is the exact solution that satisﬁes the integral equation. It is worth noting that we did not use the Adomian polynomials. This is due to the fact the modiﬁed decomposition method and the noise terms phenomenon accelerate the convergence of the solution. Example 13.11 We consider the nonlinear Volterra integral equation x 1 1 (x − t)u2 (t)dt. (13.73) u(x) = sin x + sin2 x − x2 + 4 4 0 Substituting the series (13.61) and the Adomian polynomials (13.62) into the left side and the right side of (13.73) respectively we ﬁnd x ∞ ∞ 1 1 un (x) = sin x + sin2 x − x2 + (x − t) An (t)dt, (13.74) 4 4 0 n=0 n=0 where An are the Adomian polynomials for u2 (x) as shown above. Using the modiﬁed decomposition method we set 1 u0 (x) = sin x + sin2 x, 4 x 1 2 (x − t)A0 (t)dt u1 (x) = − x + 4 0 x (13.75) 1 2 1 =− x + (x − t)u20 (t)dt = − sin2 x + · · · , 4 4 0 x (x − t)Ak (t)dt, k 1. uk+1 (x) = 0

13.3 Nonlinear Volterra Integral Equations of the Second Kind

403

We can observe the appearance of the noise terms 14 sin2 x and − 41 sin2 x between u0 (x) and u1 (x) respectively. By canceling the noise term 14 sin2 x from u0 (x), we can show that u(x) = sin x, is the exact solution that satisﬁes the integral equation.

(13.76)

Example 13.12 We consider the nonlinear Volterra integral equation x 1 (x − t)u2 (t)dt. (13.77) u(x) = sec x − tan x − x2 + 2 0 Substituting the series (13.61) and the Adomian polynomials (13.62) into the left side and the right side of (13.77) respectively gives x ∞ ∞ 1 un (x) = sec x − tan x − x2 + (x − t) An (t)dt, (13.78) 2 0 n=0 n=0 where An are the Adomian polynomials for u2 (x) as shown above. Using the modiﬁed decomposition method we set u0 (x) = sec x, x 1 u1 (x) = − tan x − x2 + (x − t)A0 (t)dt 4 (13.79) 0 x 1 (x − t)u20 (t)dt = 0. = − tan x − x2 + 2 0 The other components uk , k 2 vanish as a result. Consequently, the exact solution is given by u(x) = sec x. (13.80)

Exercises 13.3.3 Solve the following nonlinear Volterra integral equations of the second kind by using the Adomian decomposition method, the modiﬁed Adomian decomposition method, or by using the noise terms phenomenon x 1. u(x) = x − u2 (t)dt 0

1 1 6 1 x + 2. u(x) = 1 + x2 − x4 − 2 6 30 3. u(x) = 1 + x2 +

x 0

(x − t)u2 (t)dt

1 4 1 6 1 8 x + x + x + 6 10 42

1 2 2 x − x3 − x4 − x5 + 2 5 x ex−t eu(t) dt 5. u(x) = x(1 − ex ) +

4. u(x) = 1 + 2x −

0

x

(xt2 − x2 t)u2 (t)dt

0

x

0

(x − t)u3 (t)dt

404

13 Nonlinear Volterra Integral Equations x 1 1 1 6. u(x) = sin x + cos x + sin 2x − x − x2 + (x − t)u2 (t)dt 4 2 2 0 x 2 2 2 7. u(x) = 1 + x2 − xex + ex −t −1 eu(t) dt 8. u(x) = ex − xex +

0

x

0

ex−2t u2 (t)dt

x 5 1 1 (x − t)u2 (t)dt cosh2 x + − 2x − x2 + 4 4 4 0 x 2 1 1 10. u(x) = sin x + cos x − cos2 x − + sin(x − t)u2 (t)dt 3 3 3 0 x 1 1 3 2 11. u(x) = sin x − cos2 x + sin 2x − ex + + ex−t u(t)dt 5 5 5 5 0 x 3 1 12. u(x) = cos x − sin 2x − sinh x + cosh(x − t)u2 (t)dt 5 5 0 9. u(x) = sinh x −

13.4 Nonlinear Volterra Integral Equations of the First Kind The standard form of the nonlinear Volterra integral equation of the ﬁrst kind is given by x K(x, t)F (u(t))dt, (13.81) f (x) = 0

where the kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x). Recall that the unknown function u(x) occurs only inside the integral sign for the Volterra integral equation of the ﬁrst kind. The linear Volterra integral equation of the ﬁrst kind is presented in Section 3.3 where three main methods were used for handling this kind of equations. To determine a solution for the nonlinear Volterra integral equation of the ﬁrst kind (13.81), we ﬁrst convert it to a linear Volterra integral equation of the ﬁrst kind of the form x

f (x) =

K(x, t)v(t)dt,

(13.82)

0

by using the transformation This in turn means that

v(x) = F (u(x)).

(13.83)

u(x) = F −1 (v(x)).

(13.84)

It is worth noting that the Volterra integral equation of the ﬁrst kind (13.82) can be solved by any method that was studied in Section 3.3. However, in this section we will handle Eq. (13.82) by the Laplace transform method and the

13.4 Nonlinear Volterra Integral Equations of the First Kind

405

conversion to Volterra integral equation of the second kind. Other methods can be found in the literature and in this text as well.

13.4.1 The Laplace Transform Method The Laplace transform method is a powerful technique that we used before for solving Volterra integral equations of the ﬁrst and the second kinds. We assume that the kernel K(x, t) is a diﬀerence kernel. Taking the Laplace transforms of both sides of (13.82) gives L{f (x)} = L{K(x − t)} × L{v(x)}, (13.85) so that V (s) =

F (s) , K(s)

(13.86)

where F (s) = L{f (x)}, K(s) = L{K(x)}, V (s) = L{v(x)}.

(13.87)

Taking the inverse Laplace transform of both sides of (13.86) gives v(x). The solution u(x) is obtained by using (13.84). It is obvious that the Laplace transform method works eﬀectively provided that F (s) = 0. (13.88) lim s→∞ K(s) The Laplace transform method will be used for studying the following nonlinear Volterra integral equations of the ﬁrst kind. Example 13.13 Solve the nonlinear Volterra integral equation of the ﬁrst kind by using the Laplace transform method x 1 2x 1 1 (x − t)u2 (t)dt. (13.89) e − x− = 4 2 4 0 We ﬁrst set v(x) = u2 (x), u(x) = ± v(x), (13.90) to carry out (13.89) into x 1 1 2x 1 e − x− = (x − t)v(t)dt. 4 2 4 0 Taking the Laplace transform of both sides of (13.91) yields 1 1 1 1 − 2− = 2 V (s), 4(s − 2) 2s 4s s or equivalently 1 , V (s) = s−2 where

(13.91)

(13.92)

(13.93)

406

13 Nonlinear Volterra Integral Equations

V (s) = L{v(x)}.

(13.94)

Taking the inverse Laplace transform of both sides of (13.93) gives v(x) = e2x .

(13.95)

The exact solutions are therefore given by u(x) = ±ex .

(13.96)

It is worth noting that two solutions were obtained because Eq. (13.89) is a nonlinear equation, and the solution may not be unique. Example 13.14 Solve the nonlinear Volterra integral equation of the ﬁrst kind by using the Laplace transform method x 1 1 sin x − x cos x = sin(x − t) sin u(t)dt. (13.97) 2 2 0 We ﬁrst set (13.98) v(x) = sin u(x), u(x) = sin−1 v(x), to carry out (13.97) into x 1 1 sin(x − t)v(t)dt. (13.99) sin x − x cos x = 2 2 0 Taking the Laplace transform of both sides of (13.99) yields 1 1 s2 − 1 = 2 − V (s), (13.100) 2 2 2 2(s + 1) 2(s + 1) s +1 or equivalently 1 . (13.101) V (s) = 2 s +1 Taking the inverse Laplace transform of both sides of the last equation gives v(x) = sin x. (13.102) The exact solution is therefore given by u(x) = x.

(13.103)

Example 13.15 Solve the nonlinear Volterra integral equation of the ﬁrst kind by using the Laplace transform method x 2 1 1 sinh(x − t)u2 (t)dt. (13.104) + cosh 2x − cosh x = 2 6 3 0 We ﬁrst set v(x) = u2 (x), u(x) = ± v(x), (13.105) to carry out (13.104) into x 2 1 1 + cosh 2x − cosh x = sinh(x − t)v(t)dt. 2 6 3 0

(13.106)

13.4 Nonlinear Volterra Integral Equations of the First Kind

Taking the Laplace transform of both sides of (13.106) yields 1 s 2s 1 + − = 2 V (s), 2s 6(s2 − 4) 3(s2 − 1) s −1 or equivalently s 1 . V (s) = − + 2 2s 2(s − 4)

407

(13.107)

(13.108)

Taking the inverse Laplace transform of both sides of the last equation gives v(x) = sinh2 x. The exact solutions are therefore given by

(13.109)

u(x) = ± sinh x.

(13.110)

Example 13.16 Solve the nonlinear Volterra integral equation of the ﬁrst kind by using the Laplace transform method x e2x − ex = ex−tu2 (t)dt. (13.111) 0

We ﬁrst set

v(x) = u2 (x),

to carry out (13.111) into e2x − ex =

u(x) = ±

v(x),

(13.112)

x

ex−tv(t)dt.

(13.113)

0

Taking the Laplace transform of both sides of (13.113) yields 1 . (13.114) V (s) = s−2 Taking the inverse Laplace transform of both sides of the last equation gives v(x) = e2x . The exact solutions are therefore given by

(13.115)

u(x) = ±ex .

(13.116)

Exercises 13.4.1 Use the Laplace transform method to solve the nonlinear Volterra integral equations of the ﬁrst kind: x 1 1 1 4 1. x2 + x3 + (x − t)u2 (t)dt x = 2 3 12 0 x 2 2 2. sin x + cos x − cos(2x) = cos(x − t)u2 (t)dt 3 3 0 x 1 1 ex−t u2 (t)dt 3. e4x − ex = 3 3 0 x 1 2 cos(x − t)u2 (t)dt 4. sin x − sin 2x = 3 3 0

408

13 Nonlinear Volterra Integral Equations

x 1 2 1 1 1 (x − t − 1)u2 (t)dt x − x − sin2 x + sin 2x = 4 2 4 4 0 x 1 1 1 1 5 x = 6. x2 + x3 + x4 + (x − t)u3 (t)dt 2 2 4 20 0 x 1 1 6x 1 e − x− = 7. (x − t)e2u(t) dt 36 6 36 0 x 1 1 ex−t e2u(t) dt 8. e6x − ex = 5 5 0

5.

13.4.2 Conversion to a Volterra Equation of the Second Kind Consider the nonlinear Volterra integral equation of the ﬁrst kind x K(x, t)F (u(t))dt, (13.117) f (x) = 0

where the kernel K(x, t) and the function f (x) are given real-valued functions, and u(x) is the function to be determined. In a manner parallel to our discussion before, we convert (13.117) to a linear Volterra integral equation of the ﬁrst kind of the form x

K(x, t)v(t)dt,

f (x) =

(13.118)

0

by using the transformation This in turn means that

v(x) = F (u(x)).

(13.119)

u(x) = F −1 (v(x)).

(13.120)

We next convert the linear Volterra integral equations of the ﬁrst kind (13.118) to a Volterra integral equations of the second kind. The conversion technique works eﬀectively only if K(x, x) = 0. Diﬀerentiating both sides of (13.118) with respect to x, and using Leibnitz rule, we ﬁnd x Kx (x, t)v(t)dt. (13.121) f (x) = K(x, x)v(x) + 0

Solving for v(x), provided that K(x, x) = 0, we obtain the Volterra integral equation of the second kind given by x f (x) 1 − Kx (x, t)v(t)dt. v(x) = (13.122) K(x, x) 0 K(x, x) Having converted the Volterra integral equation of the ﬁrst kind to the Volterra integral equation of the second kind, we then can use any method that was presented before. Because we solved the Volterra integral equations of the second kind by many methods, we will select distinct methods for solv-

13.4 Nonlinear Volterra Integral Equations of the First Kind

409

ing the nonlinear Volterra integral equation of the ﬁrst kind after converting it to a Volterra integral equation of the second kind. Example 13.17 Convert the nonlinear Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x 1 x −2x )= ex−t u2 (t)dt. (13.123) (e − e 3 0 We ﬁrst set v(x) = u2 (x), u(x) = ± v(x), (13.124) to carry out (13.123) into x 1 x (e − e−2x ) = ex−tv(t)dt. (13.125) 3 0 Diﬀerentiating both sides of (13.125) with respect to x by using Leibnitz rule we ﬁnd the Volterra integral equation of the second kind x 1 ex−t v(t)dt. (13.126) v(x) = (ex + 2e−2x ) − 3 0 This can be solved by many methods as presented in Chapter 3. We select the Laplace transform method to solve this equation. Taking Laplace transform of both sides of (13.126) yields 1 V (s) = . (13.127) s+2 Taking the inverse Laplace transform of both sides gives (13.128) v(x) = e−2x . The exact solutions are therefore given by u(x) = ±e−x .

(13.129)

Example 13.18 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x 1 4 1 3 x + x = (x − t + 1)u2 (t)dt. (13.130) 3 12 0 We ﬁrst set (13.131) v(x) = u2 (x), u(x) = ± v(x), to carry out (13.130) into x 1 1 3 (x − t + 1)v(t)dt. (13.132) x + x4 = 3 12 0 Diﬀerentiating both sides of (13.132) with respect to x by using Leibnitz rule we ﬁnd the Volterra integral equation of the second kind x 1 v(x) = x2 + x3 − v(t)dt. (13.133) 3 0

410

13 Nonlinear Volterra Integral Equations

We will select the modiﬁed decomposition method to solve this equation. Using the recursive relation x 1 3 2 v0 (x) = x , v1 (x) = x − v0 (t)dt = 0. (13.134) 3 0 The exact solutions are therefore given by u(x) = ±x.

(13.135)

Example 13.19 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x 3 3 2x 1 e − x− = (x − t + 1)u2 (t)dt. (13.136) 4 2 4 0 We ﬁrst set (13.137) v(x) = u2 (x), u(x) = ± v(x), to carry out (13.136) into x 3 2x 1 3 (x − t + 1)v(t)dt. e − x− = 4 2 4 0 Diﬀerentiating both sides of (13.138) and proceeding as before we x 3 1 v(x) = e2x − − v(t)dt. 2 2 0 We will select the modiﬁed decomposition method to solve this Using the recursive relation x 1 1 v0 (x) = e2x , v1 (x) = e2x − − v0 (t)dt = 0. 2 2 0 The exact solutions are therefore given by u(x) = ±ex .

(13.138) ﬁnd (13.139) equation. (13.140)

(13.141)

Example 13.20 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x 1 1 sinh x + sinh 2x = cosh(x − t)u2 (t)dt. (13.142) 3 3 0 This equation is equivalent to x 1 1 cosh(x − t)v(t)dt, (13.143) sinh x + sinh 2x = 3 3 0 upon setting u2 (x) = v(x). Diﬀerentiating both sides of (13.143) and using Leibnitz rule we obtain x 2 1 sinh(x − t)v(t)dt. (13.144) v(x) = cosh x + cosh(2x) − 3 3 0 For this problem, we select the successive approximations method. We select v0 (x) = 1. Consequently, we obtain

13.5 Systems of Nonlinear Volterra Integral Equations

411

v0 (x) = 1, 2 2 v1 (x) = − cosh x + cosh 2x + 1, 3 3 4 1 4 v2 (x) = − cosh x + cosh 2x + x sinh x, (13.145) 9 9 3 14 11 1 2 14 cosh 2x + x sinh x − x cosh x + 1 v3 (x) = − cosh x + 27 27 36 12 1 4 2 6 2 = 1 + x + x + x + ··· . 3 45 The last approximation is a Taylor series for cosh2 x. This in turn gives (13.146) v(x) = cosh2 x. Consequently, the exact solution of the integral equation is given by u(x) = ± cosh x.

(13.147)

Exercises 13.4.2 Convert the nonlinear Volterra integral equations of the ﬁrst kind to the second kind linear Volterra integral equation, and solve the resulting equation: x 1 1 1 (x − t)u2 (t)dt 1. − x + x2 + sin 2x = 2 2 4 0 x 2 2 2. sin x + cos x − cos 2x = cos(x − t)u2 (t)dt 3 3 0 x 2 1 2 (x − t)u3 (t)dt 3. x − sin x − sin3 x = 3 3 9 0 x 2 1 4. sin x − sin 2x = cos(x − t)u2 (t)dt 3 3 0 x 1 1 2 5 5. x3 − x4 + (x − t)2 u2 (t)dt x = 3 3 15 0 x 1 1 1 6. − − x2 + cosh2 x = (x − t)u2 (t)dt 4 4 4 0 x 1 1 6x 1 e − x− = 7. (x − t)e2u(t) dt 36 6 36 0 x 1 1 ex−t e2u(t) dt 8. e6x − ex = 5 5 0

13.5 Systems of Nonlinear Volterra Integral Equations In this section, systems of nonlinear Volterra integral equations of the ﬁrst kind and the second kind will be studied. Many numerical and analytical methods are usually are used to handle the two kinds of systems. However, in this section we will concern ourselves with the Adomian decomposition

412

13 Nonlinear Volterra Integral Equations

method, and the successive approximations method to handle the nonlinear Volterra integral equations of the ﬁrst kind and the second kind.

13.5.1 Systems of Nonlinear Volterra Integral Equations of the Second Kind Systems of nonlinear Volterra integral equations of the second kind are given by x % 1 (x, t)F%1 (v(t)) dt, u(x) = f1 (x) + K1 (x, t)F1 (u(t)) + K 0 (13.148) x % 2 (x, t)F%2 (v(t)) dt. K2 (x, t)F2 (u(t)) + K v(x) = f2 (x) + 0

The unknown functions u(x) and v(x), that will be determined, occur inside % i (x, t), and the funcand outside the integral sign. The kernels Ki (x, t) and K tion fi (x) are given real-valued functions, for i = 1, 2. The functions Fi and F%i , for i = 1, 2 are nonlinear functions of u(x) and v(x). The Adomian decomposition method, as presented before, decomposes each solution as an inﬁnite sum of components, where these components are determined recurrently. This method can be used in its standard form, or combined with the noise terms phenomenon. Moreover, the modiﬁed decomposition method will be used wherever it is appropriate. It is interesting to point out that the nonlinear functions Fi and F%i , for i = 1, 2, should be replaced by the Adomian polynomials An deﬁned by A0 = F (u0 ), A1 = u1 F (u0 ), 1 A2 = u2 F (u0 ) + u21 F (u0 ), 2! 1 (13.149) A3 = u3 F (u0 ) + u1 u2 F (u0 ) + u31 F (u0 ), 3! 1 2 1 u2 + u1 u3 F (u0 ) + u21 u2 F (u0 ) A4 = u4 F (u0 ) + 2! 2! 1 + u41 F (iv) (u0 ). 4! As stated before, the successive substitutions method, that was used in this chapter and in other chapters as well, will be also used to study the nonlinear systems of the two kinds. Example 13.21 Use the Adomian decomposition method to solve the following system of nonlinear Volterra integral equations

13.5 Systems of Nonlinear Volterra Integral Equations

2 u(x) = x − x3 + 3

x

413

u2 (t) + v(t) dt,

0

(13.150) x 1 v(x) = x2 − x4 + u(t)v(t)dt. 4 0 The Adomian decomposition method suggests that the linear terms u(x) and v(x) be expressed by an inﬁnite series of components ∞ ∞ un (x), v(x) = vn (x), u(x) = (13.151) n=0

n=0

where un (x) and vn (x), n 0 are the components of u(x) and v(x) that will be elegantly determined in a recursive manner. The nonlinear terms u2 (t) and u(t)v(t) are given by the Adomian polynomials An and Bn : A0 = u20 , A1 = 2u0 u1 , A2 = 2u0 u2 + u21 (13.152) and B0 = u0 v0 (t), B1 = v0 (t)u1 (t) + u0 (t)v1 (t), B2 = v0 (t)u2 (t) + u0 (t)v2 (t) + u1 (t)v1 (t),

(13.153)

respectively. Substituting the previous assumptions for linear and nonlinear terms into (13.150), and following Adomian analysis, the system (13.150) is transformed into a set of recursive relations given by x 2 3 (Ak (t) + vk (t)) dt, k 0, (13.154) u0 (x) = x − x , uk+1 (x) = 3 0 and x 1 v0 (x) = x2 − x4 , vk+1 (x) = Bk (t)dt, k 0. (13.155) 4 0 This in turn gives 2 2 19 4 (13.156) u0 (x) = x − x3 , u1 (x) = x3 − x5 + x7 , 3 3 60 63 and 1 1 11 1 (13.157) v0 (x) = x2 − x4 , v1 (x) = x4 − x6 + x8 . 4 4 72 48 It is obvious that the noise terms ∓ 23 x3 appear between u0 (x) and u1 (x). Moreover, the noise terms ∓ 14 x4 appear between v0 (x) and v1 (x). By canceling these noise terms from u0 (x) and v0 (x), the non-canceled terms of u0 (x) and v0 (x) give the exact solutions (u(x), v(x)) = (x, x2 ), that satisfy the given system (13.150).

(13.158)

Example 13.22 Use the Adomian decomposition method to solve the following system of nonlinear Volterra integral equations

414

13 Nonlinear Volterra Integral Equations

1 u(x) = cos x − x2 + 2

x 0

(x − t)(u2 (t) + v 2 (t))dt,

(13.159) x 1 sin2 x + (x − t)(u2 (t) − v 2 (t))dt. 2 0 For this example, we will use the modiﬁed decomposition method, therefore we set the recursive relation u0 (x) = cos x, x (13.160) 1 (x − t)(Ak (t) + Ck (t))dt, k 0, uk+1 (x) = − x2 + 2 0 and v0 (x) = sin x, x (13.161) 1 2 (x − t)(Ak (t) − Ck (t))dt, k 0, vk+1 (x) = − sin x + 2 0 where An (t) and Cn (t) are the Adomian polynomials for u2 (t) and v 2 (t) respectively. This in turn gives u0 (x) = cos x, u1 (x) = 0, uk+1 (x) = 0, k 1, (13.162) v(x) = sin x −

and v0 (x) = sin x,

v1 (x) = 0,

k 1.

vk+1 (x) = 0,

(13.163)

This gives the exact solutions (u(x), v(x)) = (cos x, sin x).

(13.164)

Example 13.23 Use the successive approximations method to solve the following system of nonlinear Volterra integral equations x 2

xu (t) − v 2 (t) dt, u(x) = cos x + sin x + (1 + x) cos2 x − (1 + x2 ) + v(x) = cos x − sin x + (1 − x) cos2 x − (1 + x2 ) +

0 x

u2 (t) + xv 2 (t) dt.

0

(13.165) To use the successive approximations method, we ﬁrst select the zeroth approximations u0 (x) and v0 (x) by (13.166) u0 (x) = v0 (x) = 1. The method of successive approximations admits the use of the iteration formulas un+1 (x) = cos x + sin x + (1 + x) cos2 x − (1 + x2 ) x 2

+ xun (t) − vn2 (t) dt, 0 (13.167) vn+1 (x) = cos x − sin x + (1 − x) cos2 x − (1 + x2 ) x

2 un (t) + xvn2 (t) dt, n 0. + 0

13.5 Systems of Nonlinear Volterra Integral Equations

415

Substituting (13.166) into (13.167) we obtain the series u0 (x) = 1, v0 (x) = 1, 3 7 3 41 5 x + ··· , u1 (x) = 1 + x − x2 − x3 + x4 + 2 6 8 120 3 7 3 41 5 v1 (x) = 1 − x − x2 + x3 + x4 − x + ··· , 2 6 8 120 1 1 13 9 u2 (x) = 1 + x − x2 − x3 − x4 − x5 + · · · , 2 2 8 8 1 1 13 9 v2 (x) = 1 − x − x2 − x3 − x4 + x5 + · · · , 2 2 8 8 1 2 1 3 3 4 89 5 (13.168) x + ··· , u3 (x) = 1 + x − x − x + x + 2 6 8 120 1 1 3 89 5 x + ··· , v3 (x) = 1 − x − x2 + x3 + x4 − 2 6 8 120 1 1 1 1 u4 (x) = 1 + x − x2 − x3 + x4 − x5 + · · · , 2 6 24 8 1 2 1 3 1 4 1 5 v4 (x) = 1 − x − x + x + x + x + · · · , 2 6 24 8 1 2 1 3 1 4 1 u5 (x) = 1 + x − x − x + x + x5 + · · · , 2! 3! 4! 5! 1 1 1 1 v5 (x) = 1 − x − x2 + x3 + x4 − x5 + · · · . 2! 3! 4! 5! This means that the series solutions are given by 1 1 1 1 u(x) = 1 − x2 + x4 + · · · + x − x3 + x5 + · · · , 2! 4! 3! 5! (13.169) 1 4 1 3 1 5 1 2 v(x) = 1 − x + x + · · · − x − x + x + · · · . 2! 4! 3! 5! Consequently, the closed form solutions u(x) and v(x) are given by (u(x), v(x)) = (cos x + sin x, cos x − sin x), (13.170) obtained upon using the Taylor series for cos x and sin x. Notice that this system can be solved easily by using the modiﬁed decomposition method, where we can select u0 (x) = cos x + sin x and v0 (x) = cos x − sin x. Example 13.24 Use the successive approximations method to solve the following system of nonlinear Volterra integral equations x 1 (x − t)(u2 (t) − v 2 (t))dt, u(x) = ex + x − sinh 2x + 2 0 (13.171) x 2 −x x v(x) = e + x − xe + xu (t)v(t)dt. 0

416

13 Nonlinear Volterra Integral Equations

To use the successive approximations method, we ﬁrst select the zeroth approximations u0 (x) and v0 (x) by u0 (x) = v0 (x) = 1. (13.172) The method of successive approximations admits the use of the iteration formulas x 1 x (x − t)(u2n (t) − vn2 (t))dt, un+1 (x) = e + x − sinh(2x) + 2 0 (13.173) x 2 −x x vn+1 (x) = e + x − xe + xun (t)vn (t)dt, n 0. 0

Substituting (13.172) into (13.173) we obtain the series u0 (x) = 1, v0 (x) = 1, 1 1 1 u1 (x) = 1 + x + x2 − x3 + x4 + · · · , 2 2 24 1 2 1 (13.174) v1 (x) = 1 − x + x2 − x3 − x4 + · · · , 2 3 8 1 1 1 u2 (x) = 1 + x + x2 + x3 + x4 + · · · , 2! 3! 4! 1 2 1 3 1 v2 (x) = 1 − x + x − x + x4 − · · · , 2! 3! 4! and so on. Consequently, the solutions u(x) and v(x) are given by (u(x), v(x)) = (ex , e−x ).

(13.175)

Exercises 13.5.1 In Exercises 1–4, use the Adomian decomposition method to solve the systems of nonlinear Volterra integral equations ⎧ x

1 6 1 5 ⎪ ⎪ (x − t)2 u2 (t) − (x − t)v 2 (t) dt ⎪ ⎨ u(x) = x − 30 x + 30 x + 0 1. x ⎪

1 6 1 7 ⎪ ⎪ x − x + (x − t)3 u2 (t) + (x − t)2 v 2 (t) dt ⎩ v(x) = x2 − 60 105 0 ⎧ x

1 ⎪ 2 3 7 ⎪ xtu2 (t) + xtv 2 (t) dt ⎪ ⎨ u(x) = 1 + x − x − 3 x + 0 2. x ⎪

1 ⎪ ⎪ (x − t)u2 (t) − (x − t)v 2 (t) dt ⎩ v(x) = 1 − x2 − x4 + 3 0 ⎧ x

⎪ x x x ⎪ u(x) = 1 + e − 4x(1 − e + xe ) + xtu2 (t) − xtv 2 (t) dt ⎪ ⎨ 0 3. x ⎪

x−t 2 ⎪ ⎪ u (t) − ex−t v 2 (t) dt e ⎩ v(x) = 1 − ex − 4xex + 0

13.5 Systems of Nonlinear Volterra Integral Equations ⎧ x

2 ⎪ x ⎪ − sinh(2x) + u (t) + v 2 (t) dt u(x) = e ⎪ ⎨ 0 4. x ⎪ 2

⎪ ⎪ u (t) − v 2 (t) dt ⎩ v(x) = e−x + 1 − cosh(2x) +

417

0

In Exercises 5–8, use the successive approximations method to solve the systems of nonlinear Volterra integral equations ⎧ x ⎪ ⎪ u(x) = cos x − sin x + cos(x − t)(u2 (t) + v 2 (t))dt ⎪ ⎨ 5.

0

⎪ ⎪ ⎪ ⎩ v(x) = sin x + cos x − x +

x

0

sin(x − t)(u2 (t) + v 2 (t))dt

⎧ x 1 ⎪ ⎪ (u2 (t) + v 2 (t))dt sin(2x) + u(x) = 1 + sin x − 3x + ⎪ ⎨ 2 0 6. x ⎪ 3 2 1 ⎪ 2x+ ⎪ + (x − t)(u2 (t) + v 2 (t))dt v(x) = 1 − sin x − x sin ⎩ 2 2 0 ⎧ x ⎪ ⎪ u(x) = cosh x − x + (u2 (t) − v 2 (t))dt ⎪ ⎨ 0

7.

x ⎪ 1 ⎪ ⎪ (x − t)(u2 (t) + v 2 (t))dt ⎩ v(x) = sinh x − sinh2 x + 2 0 ⎧ x ⎪ ⎪ u(x) = 1 + cosh x − 4 sinh x + (u2 (t) − v 2 (t))dt ⎪ ⎨

8.

⎪ 1 ⎪ ⎪ ⎩ v(x) = 1 − cosh x − sinh(2x) − 3x + 2

0

x

0

(u2 (t) + v 2 (t))dt

13.5.2 Systems of Nonlinear Volterra Integral Equations of the First Kind In this section, we will study a speciﬁc case of the systems of nonlinear Volterra integral equations of the ﬁrst kind given by x % 1 (x, t)F%1 (v(t)) dt, f1 (x) = K1 (x, t)u(t) + K 0 (13.176) x % K2 (x, t)F2 (u(t)) + K2 (x, t)v(t) dt, f2 (x) = 0

% i (x, t), and the functions fi (x) are given where the kernels Ki (x, t) and K real-valued functions, and u(x), and v(x) are the unknown functions that will be determined. Recall that the unknown functions u(x) and v(x) appear inside the integral sign for the Volterra integral equations of the ﬁrst kind. We ﬁrst need to convert this system to a system of nonlinear Volterra integral equation of the second kind. This can be achieved by diﬀerentiating both sides of each part of the system. The conversion technique works eﬀectively by using Leibnitz rule that was presented in section 1.3. Diﬀerentiating both sides of each equation in (13.176), and using Leibnitz rule, we obtain

418

13 Nonlinear Volterra Integral Equations

% 1 (x, x)F%1 (v(x)) f1 (x) = K1 (x, x)u(x) + K x % 1x (x, t)F%1 (v(t)) dt, + K1x (x, t)u(t) + K 0

f2 (x)

% 2 (x, x)v(x) = K2 (x, x)F2 (u(x)) + K x % 2x (x, t)v(t) dt, + K2x (x, t)F2 (u(t)) + K

(13.177)

0

that can be rewritten as % 1 (x, x)F%1 (v(x)) f (x) − K u(x) = 1 K1 (x, x) x 1 % 1x (x, t)F%1 (v(t)) dt, − K1x (x, t)u(t) + K K1 (x, x) 0

(13.178) f2 (x) − K2 (x, x)F2 (u(x)) v(x) = % 2 (x, x) K x 1 % 2x (x, t)v(t) dt. − K2x (x, t)F2 (u(t)) + K % 2 (x, x) 0 K It is obvious that the last system is a system of nonlinear Volterra integral equations of the second kind. This system can be handled by many distinct methods. However, in this section we will use the Adomian decomposition method and the successive approximations method for handling the resulting system of nonlinear Volterra integral equations of the second kind. Notice that other methods can be used as well. The Adomian decomposition method and the successive approximations method were introduced before, hence we skip details. It is important to present the following two remarks:

% 2 (x, x) = 0 for the system to be 1. It is necessary that K1 (x, x) = 0 and K reduced to a system of Volterra integral equations of the second kind. % 2 (x, x) = 0, then we diﬀerentiate again. 2. If K1 (x, x) = 0 and K In the following, we will examine four examples, where Adomian method will be used in the ﬁrst two examples, and the successive approximations method will be used for the other two examples. Example 13.25 Solve the system of nonlinear Volterra integral equations of the ﬁrst kind by using the Adomian method x

1 1 1 3 (x − t + 1)u(t) + (x − t)v 2 (t) dt, x + x4 + x8 = 3 12 56 0 (13.179) x

1 1 1 4 x + x5 + x6 = (x − t)u2 (t) + (x − t + 1)v(t) dt. 4 20 30 0 Diﬀerentiating both sides of each equation, and using Leibnitz rule, we ﬁnd

13.5 Systems of Nonlinear Volterra Integral Equations

1 1 x2 + x3 + x7 = u(x) + 3 7

x

419

(u(t) + v2 (t))dt,

0 x

(13.180) 2

1 1 x3 + x4 + x5 = v(x) + u (t) + v(t) dt. 4 5 0 This system can be rewritten as x

1 1 u(t) + v 2 (t) dt, u(x) = x2 + x3 + x7 − 3 7 0 (13.181) x 2

1 1 u (t) + v(t) dt. v(x) = x3 + x4 + x5 − 4 5 0 Using the standard Adomian decomposition method, we set the recurrence relations 1 1 u0 (x) = x2 + x3 + x7 , 3 7 1 1 v0 (x) = x3 + x4 + x5 , 4 5 x (13.182)

1 1 u0 (t) + v02 (t) dt = − x3 − x7 + · · · , u1 (x) = − 3 7 0 x 2

1 1 v1 (x) = − u0 (t) + v0 (t) dt = − x4 − x5 + · · · . 4 5 0 It is obvious that two noise terms appear between u0 (x) and u1 (x), and two other noise terms between v0 (x) and v1 (x). Canceling the noise terms in u0 (x) and v0 (x) gives the exact solutions by (u(x), v(x)) = (x2 , x3 ). (13.183) Example 13.26 Solve the system of nonlinear Volterra integral equations of the ﬁrst kind by using the modiﬁed Adomian method 1 1 1 1 1 sin x + sin(2x) − sin2 x + x2 − x − 4 4 4 2 4 x

= u(t) + (x − t − 1)v 2 (t) dt, 0 (13.184) 1 2 5 1 2 − cos x − sin x − cos x + x + x + 4 4 4 x

2 = (x − t)u (t) + (x − t + 1)v(t) dt. 0

Diﬀerentiating both sides of each equation, and using Leibnitz rule, we ﬁnd x 1 1 1 1 v2 (t)dt, u(x) = cos x + cos(2x)− sin(2x) + x − − v 2 (x) − 2 4 2 2 0 (13.185) x

2 1 1 v(x) = sin x − cos x + sin(2x) + x + 1 − u (t) + v(t) dt. 4 2 0

420

13 Nonlinear Volterra Integral Equations

To handle this example we select the modiﬁed decomposition method, hence we set the recurrence relations u0 (x) = cos x, v0 (x) = sin x, x 1 1 1 1 v02 (t)dt = 0, u1 (x) = cos(2x) − sin(2x)+ x − −v02 (x) − 2 4 2 2 (13.186) 0 x

1 1 v1 (x) = − cos x + sin(2x) + x + 1 − u20 (t) + v0 (t) dt = 0. 4 2 0 This in turn gives the exact solutions by (u(x), v(x)) = (cos x, sin x).

(13.187)

Example 13.27 Solve the system of nonlinear Volterra integral equations of the ﬁrst kind by using the successive approximations method x

9 1 −2x 1 x (x − t + 1)u(t) + (x − t)v 2 (t) dt, − x− = 2e + e 4 2 4 0 (13.188) x

1 2x 1 1 2 (x − t)u (t) + (x − t + 1)v(t) dt. e + x− = 4 2 4 0 Diﬀerentiating both sides of each equation, and using Leibnitz rule, we ﬁnd x

1 −2x 1 x u(t) + v 2 (t) dt, u(x) = 2e − e − − 2 2 0 (13.189) x

2 1 2x 1 u (t) + v(t) dt. v(x) = e + − 2 2 0 To use the successive approximations method, we ﬁrst select the zeroth approximations u0 (x) and v0 (x) by (13.190) u0 (x) = v0 (x) = 1. The method of successive approximations admits the use of the iteration formulas x

1 1 un+1 (x) = 2ex − e−2x − − un (t) + vn2 (t) dt, 2 2 0 (13.191) x

1 1 vn+1 (x) = e2x + − u2n (t) + vn (t) dt, n 0. 2 2 0 Substituting (13.190) into (13.191) we obtain the series u0 (x) = 1, v0 (x) = 1, 1 u1 (x) = 1 + x + x3 − x4 + · · · , 4 2 1 2 v1 (x) = 1 − x + x + x3 + x4 + · · · , 3 3 1 2 1 4 u2 (x) = 1 + x + x − x + · · · , 2 3

13.5 Systems of Nonlinear Volterra Integral Equations

1 1 v2 (x) = 1 − x + x2 − x4 + · · · , 2 3

421

(13.192)

1 1 u3 (x) = 1 + x + x2 + x3 + · · · , 2 6 1 1 1 v3 (x) = 1 − x + x2 − x3 + x4 + · · · , 2 6 12 u4 (x) = 1 + x +

1 2 1 1 x + x3 + x4 + · · · , 2! 3! 4!

1 2 1 1 x − x3 + x4 − · · · . 2! 3! 4! Consequently, the solutions u(x) and v(x) are given by v4 (x) = 1 − x +

(u(x), v(x)) = (ex , e−x ).

(13.193)

Example 13.28 Solve the system of nonlinear Volterra integral equations of the ﬁrst kind by using the successive approximations method 1 1 1 −2 cos x + sin(2x) + x2 + x + 2 4 2 2 x

(x − t + 1)u(t) + (x − t)v 2 (t) dt, = 0 (13.194) 1 1 1 2 sin x − sin(2x) + x2 − x 4 2 2 x

= (x − t)u2 (t) + (x − t + 1)v(t) dt. 0

Diﬀerentiating both sides of each equation, and using Leibnitz rule, we ﬁnd x

1 1 u(x) = 2 sin x + cos(2x) + x + − u(t) + v 2 (t) dt, 2 2 0 (13.195) x

2 1 1 v(x) = 2 cos x − cos(2x) + x − − u (t) + v(t) dt. 2 2 0 We select the zeroth approximations u0 (x) and v0 (x) by u0 (x) = v0 (x) = 1. We next use the iteration formulas x

1 1 un (t) + vn2 (t) dt, un+1 (x) = 2 sin x + cos(2x) + x + − 2 2 0 x

2 1 1 un(t) + vn (t) dt, n 0. vn+1 (x) = 2 cos x − cos(2x) + x − − 2 2 0 (13.196) Substituting the zeroth selections into (13.196) we obtain the series

422

13 Nonlinear Volterra Integral Equations

u0 (x) = 1,

v0 (x) = 1, 1 1 1 u1 (x) = 1 + x − x2 − x3 + x4 + x5 + · · · , 3 3 60 1 v1 (x) = 1 − x − x4 + · · · , 4 1 1 5 1 u2 (x) = 1 + x − x2 − x3 + x4 + x5 + · · · , 2 3 12 20 1 2 1 3 5 4 3 v2 (x) = 1 − x − x + x + x − x5 + · · · , 2 3 12 20 1 1 3 u3 (x) = 1 + x − x2 − x3 − x5 + · · · , (13.197) 2 6 20 1 2 1 3 1 4 1 5 v3 (x) = 1 − x − x + x + x − x + · · · , 2 6 12 6 1 2 1 3 1 4 u4 (x) = 1 + x − x − x + x + · · · , 2 6 24 1 2 1 3 1 v4 (x) = 1 − x − x + x + x4 + · · · , 2 6 24 1 1 1 1 u5 (x) = 1 + x − x2 − x3 + x4 + x5 + · · · , 2! 3! 4! 5! 1 2 1 3 1 4 1 v5 (x) = 1 − x − x + x + x − x5 + · · · . 2! 3! 4! 5! This means that the series solutions are given by 1 4 1 3 1 5 1 2 u(x) = 1 − x + x + · · · + x − x + x + · · · , 2! 4! 3! 5! (13.198) 1 2 1 4 1 3 1 5 v(x) = 1 − x + x + · · · − x − x + x + · · · . 2! 4! 3! 5! Consequently, the closed form solutions u(x) and v(x) are given by (u(x), v(x)) = (cos x + sin x, cos x − sin x). (13.199)

Exercises 13.5.2 In Exercises 1–4, use the Adomian decomposition method to solve the following systems of Volterra integral equations of the ﬁrst kind ⎧ x

1 2 1 5 1 6 ⎪ ⎪ x x x u(t) + (x − t + 1)v 2 (t) dt + + = ⎪ ⎨2 5 30 0 1. x ⎪

1 1 ⎪ ⎪ (x − t)u2 (t) + (x − t + 1)v(t) dt ⎩ x3 + x4 = 3 6 0 ⎧ x

⎪ 1 x4 + 1 x5 + 1 x12 = ⎪ (x − t + 1)u(t) + (x − t)v 2 (t) dt ⎪ ⎨4 20 132 0 2. x ⎪

1 1 1 ⎪ 6 7 8 ⎪ x + x = (x − t)u2 (t) + (x − t + 1)v(t) dt ⎩ x + 6 42 56 0

References 423 ⎧ x

1 2 3 1 ⎪ 2 ⎪ u(t) − (x − t)v 2 (t) dt ⎪ ⎨ − cos x + 4 cos x − 4 x + 4 = 0 3. x ⎪

1 1 3 ⎪ ⎪ (x − t)u2 (t) + (x − t + 1)v(t) dt ⎩ sin x − cos x + cos2 x + x2 + = 4 4 4 0 ⎧ x

1 2 1 1 ⎪ 2 ⎪ u(t) − (x − t)v 2 (t) dt ⎪ ⎨ sinh x − 4 cosh x + 4 x + 4 = 0 4. x ⎪

1 5 1 ⎪ ⎪ (x − t)u2 (t)v(t) dt ⎩ cosh x + cosh2 x + x2 − = 4 4 4 0 In Exercises 5–8, use the successive approximations method to solve the systems of Volterra integral equations of the ﬁrst kind ⎧ x

⎪ 2 ⎪ x + 2x = 1 − 2 cos x + cos (x − t + 1)u(t) + v 2 (t) dt ⎪ ⎨ 0 5. x ⎪ 2

⎪ ⎪ u (t) + (x − t + 1)v(t) dt ⎩ 1 − 2 sin x − cos2 x + 2x = 0

⎧ x

1 3 2x 1 −4x 1 ⎪ ⎪ − − e e x − = (x − t + 1)u(t) + v 2 (t) dt ⎪ ⎨4 4 2 2 0 6. x ⎪

1 1 1 ⎪ ⎪ u2 (t) + (x − t + 1)v(t) dt ⎩ e4x − e−4x + x = 4 4 2 0 ⎧ x

1 2x 1 2 1 ⎪ ⎪ e + x +x− = (x − t + 1)u(t) + v 2 (t) dt ⎪ ⎨2 2 2 0 7. x ⎪ 2

1 1 1 ⎪ ⎪ u (t) + (x − t + 1)v(t) dt ⎩ e2x + x2 + 3x − = 2 2 2 0 ⎧ x

5 2 3 1 ⎪ 2 ⎪ x + + cos x = (x − t + 1)u(t) + (x − t)v 2 (t) dt sin x − cos x + ⎪ ⎨ 4 4 4 0 8. x ⎪

1 5 5 ⎪ ⎪ (x − t)u2 (t) + (x − t + 1)v(t) dt ⎩ − sin x + cos x+ cos2 x+ x2 + 2x − = 4 4 4 0

References 1. H.T. Davis, Introduction to Nonlinear Diﬀerential and Integral Equations, Dover, Publications, New York, (1962). 2. A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 3. P. Linz, A simple approximation method for solving Volterra integro-diﬀerential equations of the ﬁrst kind, J. Inst. Math. Appl., 14 (1974) 211–215. 4. P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, (1985). 5. R.K. Miller, Nonlinear Volterra Integral Equations, W.A. Benjamin, Menlo Park, CA, (1967). 6. W.E. Olmstead and R.A. Handelsman, Asymptotic solution to a class of nonlinear Volterra integral equations, II, SIAM J. Appl. Math., 30 (1976) 180–189. 7. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).

Chapter 14

Nonlinear Volterra Integro-Diﬀerential Equations

14.1 Introduction It is well known that linear and nonlinear Volterra integral equations arise in many scientiﬁc ﬁelds such as the population dynamics, spread of epidemics, and semi-conductor devices. Volterra started working on integral equations in 1884, but his serious study began in 1896. The name integral equation was given by du Bois-Reymond in 1888. The linear Volterra integro-diﬀerential equations were presented in Chapter 5. It is our goal in this chapter to study the nonlinear Volterra integrodiﬀerential equations of the ﬁrst and the second kind. The nonlinear Volterra integro-diﬀerential equations are characterized by at least one variable limit of integration. The nonlinear Volterra integro-diﬀerential equation of the second kind reads x K(x, t)F (u(t)) dt, (14.1) u(n) (x) = f (x) + 0

and the standard form of the nonlinear Volterra integro-diﬀerential equation [1–3] of the ﬁrst kind is given by x x K1 (x, t)F (u(t))dt + K2 (x, t)u(n) (t)dt = f (x), K2 (x, x) = 0, (14.2) 0

0

where u(n) (x) is the n th derivative of u(x). For these equations, the kernels K(x, t), K1 (x, t) and K2 (x, t), and the function f (x) are given realvalued functions. The function F (u(x)) is a nonlinear function of u(x) such as u2 (x), sin(u(x)), and eu(x) .

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

426

14 Nonlinear Volterra Integro-Diﬀerential Equations

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind The linear Volterra integro-diﬀerential equation [4–6], where both diﬀerential and integral operators appear together in the same equation, has been studied in Chapter 5. In this section, we will extend the work presented in Chapter 5 to nonlinear Volterra integro-diﬀerential equation. The nonlinear Volterra integro-diﬀerential equation of the second kind reads x u(i) (x) = f (x) + K(x, t)F (u(t))dt, (14.3) 0 i

where u(i) (x) = ddxui , and F (u(x)) is a nonlinear function of u(x). Because the equation in (14.3) combines the diﬀerential operator and the integral operator, then it is necessary to deﬁne initial conditions for the determination of the particular solution u(x) of the nonlinear Volterra integro-diﬀerential equation. The nonlinear Volterra integro-diﬀerential equation [7–10] appeared after its establishment by Volterra. It appears in many physical applications such as glass-forming process, heat transfer, diﬀusion process in general, neutron diﬀusion and biological species coexisting together with increasing and decreasing rates of generating. More details about the sources where these equations arise can be found in physics, biology and books of engineering applications. In Chapter 5, we applied many methods to handle the linear Volterra integro-diﬀerential equations of the second kind. In this section we will use only some of these methods. However, the other methods presented in Chapter 5 can be used as well. In what follows we will apply the combined Laplace transform-Adomian decomposition method, the variational iteration method (VIM), and the series solution method to handle nonlinear Volterra integrodiﬀerential equations of the second kind (14.3).

14.2.1 The Combined Laplace Transform-Adomian Decomposition Method In this section we will consider the kernel K(x, t) of (14.3) as a diﬀerence kernel that depends on the diﬀerence x − t, such as ex−t, cosh(x − t), and sin(x − t). The nonlinear Volterra integro-diﬀerential equation (14.3) can thus be expressed as x u(i) (x) = f (x) + K(x − t)F (u(t)) dt. (14.4) 0

To solve the nonlinear Volterra integro-diﬀerential equations by using the Laplace transform method, it is essential to use the Laplace transforms of

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

427

the derivatives of u(x). We can easily show that L{u(i) (x)} = si L{u(x)} − si−1 u(0) − si−2 u (0) − · · · − u(i−1) (0). This simply gives L{u (x)} = sL{u(x)} − u(0) = sU (s) − u(0), L{u (x)} = s2 L{u(x)} − su(0) − u (0) = s2 U (s) − su(0) − u (0), L{u (x)} = s3 L{u(x)} − s2 u(0) − su (0) − u (0) = s3 U (s) − s2 u(0) − su (0) − u (0),

(14.5)

(14.6)

L{u(iv) (x)} = s4 L{u(x)} − s3 u(0) − s2 u (0) − su (0) − u (0) = s4 U (s) − s3 u(0) − s2 u (0) − su (0) − u (0), and so on for derivatives of higher order, where U (s) = L{u(x)}. Applying the Laplace transform to both sides of (14.4) gives si L{u(x)} − si−1 u(0) − si−2 u (0) − · · · − u(i−1) (0) (14.7) = L{f (x)} + L{K(x − t)}L{F (u(t))}, or equivalently 1 1 1 L{u(x)} = u(0) + 2 u (0) + · · · + i u(i−1) (0) s s s (14.8) 1 1 + i L{f (x)} + i L{K(x − t)}L{F (u(t))}. s s To overcome the diﬃculty of the nonlinear term F (u(x)), we apply the Adomian decomposition method for handling (14.8). To achieve this goal, we ﬁrst represent the linear term u(x) at the left side by an inﬁnite series of components given by ∞ u(x) = un (x), (14.9) n=0

where the components un (x), n 0 will be recursively determined. However, the nonlinear term F (u(x)) at the right side of (14.8) will be represented by an inﬁnite series of the Adomian polynomials An in the form n ∞ 1 dn i An (x), An = λ ui , n = 0, 1, 2, · · · , F F (u(x)) = n! dλn n=0 i=0 λ=0 (14.10) where An , n 0 can be obtained for all forms of nonlinearity. Substituting (14.9) and (14.10) into (14.8) leads to + ∞ 1 1 1 un (x) = u(0) + 2 u (0) + · · · + i u(i−1) (0) L s s s n=0 ∞ + (14.11) 1 1 + i L{f (x)} + i L{K(x − t)}L An (x) . s s n=0

428

14 Nonlinear Volterra Integro-Diﬀerential Equations

The Adomian decomposition method admits the use of the following recursive relation 1 1 1 1 L{u0 (x)} = u(0) + 2 u (0) + · · · + i u(i−1) (0) + i L{f (x)}, s s s s (14.12) 1 L{uk+1 (x)} = i L{K(x − t)}L{Ak (x)}, k 0. s The necessary conditions presented in Chapter 1 for Laplace transform method concerning the limit as s → ∞, should be satisﬁed here for a successful use of this method. Applying the inverse Laplace transform to the ﬁrst part of (14.12) gives u0 (x), that will deﬁne A0 . This in turn will lead to the complete determination of the components of uk+1 , k 0 upon using the second part of (14.12). The combined Laplace transform Adomian-decomposition method for solving nonlinear Volterra integro-diﬀerential equations of the second kind will be illustrated by studying the following examples. Example 14.1 Solve the nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method x 1 2 9 5 1 −2x −x u (x) = − x − x − 3e − e + (x − t)u2 (t) dt, u(0) = 2. (14.13) 4 2 2 4 0 Notice that the kernel K(x − t) = (x − t). Taking Laplace transform of both sides of (14.13) gives 1 2 9 5 1 −2x −x − x − x − 3e − e L{u (x)} = L +L{(x−t)∗u2 (x)}, (14.14) 4 2 2 4 so that 5 1 1 9 1 3 − − + L{u2 (x)}, (14.15) sU (s) − u(0) = − 3− 4s 2s2 s s + 1 4(s + 2) s2 or equivalently 9 1 1 5 1 3 2 − + 3 L{u2 (x)}. (14.16) U (s) = + 2 − 3 − 4 − s 4s 2s s s(s + 1) 4s(s + 2) s Substituting the series assumption for U (s) and the Adomian polynomials for u2 (x) as given above in (14.9) and (14.10) respectively, and using the recursive relation (14.12) we obtain 2 5 1 3 9 1 − , U0 (s) = + 2 − 3 − 4 − s 4s 2s s s(s + 1) 4s(s + 2) (14.17) 1 L{uk+1 (x)} = 3 L{Ak (x)}, k 0. s Recall that the Adomian polynomials for F (u(x)) = u2 (x) are given by A0 = u20 , A1 = 2u0 u1 , A2 = 2u0 u2 + u21 , (14.18) A3 = 2u0 u3 + 2u1 u2 .

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

429

Taking the inverse Laplace transform of both sides of the ﬁrst part of (14.17), and using the recursive relation (14.17) gives 1 5 5 7 5 u0 (x) = 2 − x + x2 − x3 + x4 − x + ··· , 2 6 24 120 (14.19) 2 1 1 u1 (x) = x3 − x4 + x5 + · · · . 3 6 20 Using (14.9), the series solution is therefore given by 1 1 1 1 (14.20) u(x) = 2 − x + x2 − x3 + x4 − x5 + · · · , 2! 3! 4! 5! that converges to the exact solution u(x) = 1 + e−x . (14.21) Example 14.2 Solve the nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method x 2 cos(x − t)u2 (t)dt, u(0) = 1. (14.22) u (x) = − (2 sin x + sin(2x)) + 3 0 Notice that the kernel K(x − t) = cos(x − t). Taking Laplace transform of both sides of (14.22) gives 2 L{u (x)} = L − (2 sin x + sin(2x)) + L{cos(x − t) ∗ u2 (x)}, (14.23) 3 so that 4 2 s (14.24) sU (s) − u(0) = − 2 − + 2 L{u2 (x)}, 2 3(s + 1) 3(s + 4) s + 1 or equivalently 4 2 1 1 − + 2 L{u2 (x)}. U (s) = − (14.25) 2 2 s 3s(s + 1) 3s(s + 4) s + 1 Substituting the series assumption for U (s) and the Adomian polynomials for u2 (x) as given above in (14.9) and (14.10) respectively, and using the recursive relation (14.12) we obtain 2 5 1 3 9 1 − , U0 (s) = + 2 − 3 − 4 − s 4s 2s s s(s + 1) 4s(s + 2) (14.26) 1 L{uk+1 (x)} = 3 L{Ak (x)}, k 0. s Taking the inverse Laplace transform of both sides of the ﬁrst part of (14.26), and using the recursive relation (14.26) gives 1 1 u0 (x) = 1 − x2 + x4 − x6 + · · · , 6 60 1 5 37 6 u1 (x) = x2 − x4 + x + ··· , (14.27) 2 24 720 1 4 1 x − x6 + · · · , u2 (x) = 22 20

430

14 Nonlinear Volterra Integro-Diﬀerential Equations

1 6 x + ··· . 72 Using (14.9), the series solution is therefore given by 1 1 1 u(x) = 1 − x2 + x4 − x6 + · · · , 2! 4! 6! that converges to the exact solution u(x) = cos x. u3 (x) =

(14.28) (14.29)

Example 14.3 Solve the nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method x 2 2 x 2x ex−t u2 (t)dt, u(0) = 1, u (0) = 2. u (x) = 2 + 2x + x − x e − e + 0

(14.30) Notice that the kernel K(x − t) = ex−t . Taking Laplace transform of both sides of (14.30) gives (14.31) L{u (x)} = L{2 + 2x + x2 − x2 ex − e2x } + L{ex−t ∗ u2 (x)}, so that s2 U (s) − su(0) − u (0) =

2 1 2 2 2 1 + + L{u2 (x)}, + 3 − − s s2 s (s − 1)3 s − 2 s − 1 (14.32)

or equivalently 2 2 2 2 1 1 2 1 − 2 U (s) = + 2 + 3 + 4 + 5 − 2 + 2 L{u2 (x)}. 3 s s s s s s (s − 1) s (s − 2) s (s − 1) (14.33) Proceeding as before we ﬁnd 2 1 2 2 2 2 1 U0 (s) = + 2 + 3 + 4 + 5 − 2 , − 2 3 s s s s s s (s − 1) s (s − 2) (14.34) 1 L{uk+1 (x)} = 2 L{Ak (x)}, k 0. s (s − 1) Taking the inverse Laplace transform of both sides of the ﬁrst part of (14.34), and using the recursive relation (14.34) gives 1 1 7 7 6 x + ··· , u0 (x) = 1 + 2x + x2 − x4 − x5 − 2 6 60 180 1 5 1 3 (14.35) u1 (x) = x3 + x4 + x5 + x6 + · · · , 6 24 8 80 1 6 u2 (x) = x + ··· . 360 Using (14.9), the series solution is therefore given by 1 2 1 3 1 4 1 5 1 6 u(x) = x + 1 + x + x + x + x + x + x + · · · , (14.36) 2! 3! 4! 5! 6! that converges to the exact solution

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

u(x) = x + ex .

431

(14.37)

Example 14.4 Solve the nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method x 1 sin(x − t)u2 (t)dt, (14.38) u (x) = −1 − (sin x + sin(2x)) + 2 cos x + 3 0 where u(0) = −1, u (0) = 1. Notice that the kernel K(x − t) = sin(x − t). Taking Laplace transform of both sides of (14.38) gives 1 L{u (x)} = L −1 − (sin x + sin(2x)) + 2 cos x + L{sin(x − t) ∗ u2 (x)}, 3 (14.39) so that 1 2 2s 1 1 s2 U (s)−su(0)−u (0) = − − 2 − + + L{u2 (x)}, s 3(s + 1) 3(s2 + 4) s2 + 1 s2 + 1 (14.40) so that 1 2 2 1 1 1 − + U (s) = − + 2 − 3 − 2 2 s s s 3s (s + 1) 3s2 (s2 + 4) s(s2 + 1) 1 + 2 2 (14.41) L{u2 (x)}. s (s + 1) Proceeding as before we ﬁnd 1 1 1 1 1 6 11 7 u0 (x) = −1 + x + x2 − x3 − x4 + x5 + x − x + ··· , 2 6 12 40 360 5040 1 4 1 1 6 1 7 x − x5 − x + x + ··· . (14.42) u1 (x) = 24 60 720 504 Using (14.9), the series solution is therefore given by 1 1 1 1 1 1 u(x) = x − x3 + x5 − x7 + · · · − 1 − x2 + x4 − x6 + · · · , 3! 5! 7! 2! 4! 6! (14.43) that converges to the exact solution u(x) = sin x − cos x. (14.44)

Exercises 14.2.1 Solve the following nonlinear Volterra integro-diﬀerential equations by using the combined Laplace transform-Adomian decomposition method x 17 1 9 1. u (x) = (x − t)u2 (t)dt, u(0) = 3 + x − 2x2 − 3ex − e2x + 4 2 4 0 x 11 1 3 1 4 (x − t)u2 (t)dt, u(0) = 1 − x− x + ex (5 − 2x) − e2x + 2. u (x) = − 4 2 12 4 0

432 3. 4. 5. 6.

14 Nonlinear Volterra Integro-Diﬀerential Equations x 1 1 u (x) = 1 − ex + e−2x + ex−t u2 (t)dt, u(0) = 0 3 3 0 x 5 1 1 u (x) = − + cos x − cos2 x + sin(x − t)u2 (t)dt, u(0) = 0 3 3 3 0 x 1 5 u (x) = − sin x + sin(2x) + cos(x − t)u2 (t)dt, u(0) = 0, u (0) = 1 3 3 0 x 1 1 cosh(x − t)u2 (t)dt, u (x) = cosh x − sinh x − sinh(2x) + 3 3 0 u(0) = 1, u (0) = 0

7. u (x) =

3 x 1 3x e − e + 2 2

8. u (x) = −

x

0

ex−t u3 (t)dt, u(0) = u (0) = u (0) = 1

2 5 4 − cos x + cos2 x + 3 3 3

x

0

cos(x − t)u2 (t)dt,

u(0) = u (0) = 1, u (0) = −1

14.2.2 The Variational Iteration Method The variational iteration method was used before in other chapters. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the Adomian polynomials. The standard i th order nonlinear Volterra integro-diﬀerential equation is of the form x u(i) (x) = f (x) + K(x, t)F (u(t))dt, (14.45) 0 (i)

di u dxi ,

and F (u(x)) is a nonlinear function of u(x). The initial where u (x) = conditions should be prescribed for the complete determination of the exact solution. The correction functional for the nonlinear integro-diﬀerential equation (14.45) is x

un+1 (x) = un (x) +

0

λ(ξ) u(i) n (ξ) − f (ξ) −

ξ

0

K(ξ, r)F (˜ un (r)) dr

dξ.

(14.46) The variational iteration method is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ that can be identiﬁed optimally via integration by parts and by using a restricted variation. The Lagrange multiplier λ may be a constant or a function. Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), n 0 of the solution

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

433

u(x). The zeroth approximation u0 can be any selective function. However, the initial values u(0), u (0), . . . are preferably used for the selective zeroth approximation u0 . In what follows we summarize the Lagrange multipliers as derived in Chapter 3, and the selective zeroth approximations: u + f (u(ξ), u (ξ)) = 0, λ = −1, u0 (x) = u(0), u + f (u(ξ), u (ξ), u (ξ)) = 0, λ = ξ − x, u0 (x) = u(0) + u (0)x 1 (14.47) u + f (u(ξ), u (ξ), u (ξ), u (ξ)) = 0, λ = − (ξ − x)2 , 2! 1 u0 (x) = u(0) + u (0)x + u (0)x2 , 2! and so on. Consequently, the solution is given by u(x) = lim un (x). (14.48) n→∞

The VIM will be illustrated by studying the following examples. Example 14.5 Use the variational iteration method to solve the nonlinear Volterra integrodiﬀerential equation x x x 2x ex−t u2 (t)dt, u(0) = 2. (14.49) u (x) = 1 + e − 2xe − e + 0

The correction functional for this equation is given by x t un (t) − 1 − et + 2tet + e2t − un+1 (x) = un (x) − et−r u2n(r) dr dt, 0

0

(14.50) where we used λ = −1 for ﬁrst-order integro-diﬀerential equation. We can use the initial condition to select u0 (x) = u(0) = 2. Using this selection into the correction functional gives the following successive approximations u0 (x) = 2, 1 1 3 19 5 x + ··· , u1 (x) = 2 + x + x2 − x3 − x4 − 2 2 8 120 (14.51) 1 1 1 1 u2 (x) = 2 + x + x2 + x3 + x4 − x5 + · · · , 2! 3! 4! 8 1 2 1 3 1 4 1 u3 (x) = 2 + x + x + x + x + x5 + · · · , 2! 3! 4! 5! and so on for other approximations. The VIM admits the use of u(x) = lim un (x), n→∞

that gives the exact solution by u(x) = 1 + ex .

(14.52) (14.53)

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14 Nonlinear Volterra Integro-Diﬀerential Equations

Example 14.6 Use the variational iteration method to solve the nonlinear Volterra integrodiﬀerential equation x u (x) = −x + sec x tan x − tan x + (1 + u2 (t)) dt, u(0) = 1. (14.54) 0

The correction functional for this equation is given by un+1 (x) = un (x) x t un (t) + t − sec t tan t + tan t − − (1 + u2n (r))dr dt, 0

0

(14.55) where we used λ = −1 for ﬁrst-order integro-diﬀerential equation. We can use the initial condition to select u0 (x) = u(0) = 1. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1, 1 1 1 u1 (x) = 1 + x2 + x4 + x6 + · · · , 2 8 16 (14.56) 1 2 5 4 19 6 u2 (x) = 1 + x + x + x + ··· , 2 24 240 1 2 5 4 61 6 x + ··· , u3 (x) = 1 + x + x + 2 24 720 and so on. The VIM admits the use of (14.57) u(x) = lim un (x), n→∞

that gives the exact solution by u(x) = sec x.

(14.58)

Example 14.7 Use the variational iteration method to solve the nonlinear Volterra integrodiﬀerential equation x u (x) = x + cos x − tan x + tan2 x + (sin t + u2 (t)) dt, u(0) = 0. (14.59) 0

The correction functional for this equation is given by un+1 (x) = un (x) t x (sin t + u2n (r))dr dt. un(t) − t − cos t + tan t − tan2 t − − 0

0

(14.60) We can use the initial condition to select u0 (x) = u(0) = 0. Using this selection into the correction functional gives the following successive approximations

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

435

u0 (x) = 0, 1 1 2 1 17 7 x + ··· , u1 (x) = x + x3 − x4 + x5 − x6 + 3 12 15 45 315 1 2 17 7 u2 (x) = x + x3 + x5 + x + ··· , 3 15 315 and so on. The VIM admits the use of u(x) = lim un (x), n→∞

that gives the exact solution by u(x) = tan x.

(14.61)

(14.62) (14.63)

Example 14.8 Use the variational iteration method to solve the nonlinear Volterra integrodiﬀerential equation x 1 5 u (x) = − sin x + sin(2x) + cos(x − t)u2 (t)dt, u(0) = 0, u (0) = 1. 3 3 0 (14.64) The correction functional for this equation is given by un+1 (x) = un (x) t x 1 5 2 (t − x) un (t) + sin t − sin(2t) − cos(t − r)un (r)dr dt. + 3 3 0 0 (14.65) We can use the initial condition to select u0 (x) = u(0) + xu (0) = x. Using this selection into the correction functional gives the following successive approximations u0 (x) = x, 1 1 1 7 x + ··· , u1 (x) = x − x3 + x5 + (14.66) 3! 5! 720 1 1 1 u2 (x) = x − x3 + x5 − x7 + · · · , 3! 5! 7! and so on. The VIM gives the exact solution by u(x) = sin x.

(14.67)

Exercises 14.2.2 Solve the following nonlinear Volterra integro-diﬀerential equations by using the variational iteration method x 1 1 1. u (x) = (1 − x2 ) − sin x − cos2 x + (x − t)(1 − u2 (t))dt, u(0) = 0 4 4 0 x 1 1 (x − t)(1 − u2 (t))dt, u(0) = 1 2. u (x) = − (1 − 2x + 2x2 ) − e−x + e−2x + 4 4 0 x 2 2 xte−u (t)dt, u(0) = 0 3. u (x) = 1 + u − xe−x − 2 0

436

14 Nonlinear Volterra Integro-Diﬀerential Equations x 4 1 4. u (x) = − sin x − sin(2x) + cos(x − t)u2 (t)dt, u(0) = 1 3 3 0 x 1 1 5. u (x) = − x − cos x + sin(2x) + (1 − u2 (t))dt, u(0) = 1, u (0) = 0 2 4 0 x 6. u (x) = 1 + ex − e2x + ex−t (1 + u2 (t))dt, u(0) = 1, u (0) = 1 0

x

7. u (x) = e (2 − x) − e

2x

x

+ 0

ex−t (u + u2 (t))dt, u(0) = 1, u (0) = 1

8. u (x) = −x + tan x(6 sec3 x − sec x + 1) +

x

0

(1 − u2 (t))dt,

u(0) = 1, u (0) = 0, u (0) = 1

14.2.3 The Series Solution Method The series solution method [5,7] was eﬀectively used in this text to handle integral and integro-diﬀerential equations. The method stems mainly from the Taylor series for analytic functions. A real function u(x) is called analytic if it has derivatives of all orders such that the Taylor series at any point b in its domain k u(n) (b) (x − b)n , uk (x) = (14.68) n! n=0 converges to u(x) in a neighborhood of b. For simplicity, the generic form of Taylor series at x = 0 can be written as ∞ a n xn . (14.69) u(x) = n=0

The Taylor series method, or simply the series solution method will be used in this section for solving nonlinear Volterra integro-diﬀerential equations. We will assume that the solution u(x) of the nonlinear Volterra integrodiﬀerential equation x u(n) (x) = f (x) + λ K(x, t)F (u(t))dt, u(k) (0) = k!ak , 0 k (n − 1), 0

(14.70) is analytic, and therefore possesses a Taylor series of the form given in (14.69), where the coeﬃcients an will be determined recurrently. The ﬁrst few coeﬃcients ak can be determined by using the initial conditions so that 1 1 (14.71) a0 = u(0), a1 = u (0), a2 = u (0), a3 = u (0), 2! 3! and so on. The remaining coeﬃcients ak of (14.69) will be determined by applying the series solution method to the nonlinear Volterra integro-diﬀerential equation (14.70). Substituting (14.69) into both sides of (14.70) gives

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

∞

(n) ak x

k

x

= T (f (x)) +

K(x, t)F 0

k=0

∞

437

ak t

k

dt,

(14.72)

k=0

where T (f (x)) is the Taylor series for f (x). The integro-diﬀerential equation (14.70) will be converted to a traditional integral in (14.72) where instead of integrating the unknown function F (u(x)), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integral in (14.72), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0, where some of these coeﬃcients will be used from the initial conditions. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (14.69). Example 14.9 Solve the nonlinear Volterra integro-diﬀerential equation by using the series solution method x x 2x u (x) = 1 − e + e + ex−t (1 − u2 (t))dt, u(0) = 1. (14.73) 0

Substituting u(x) by the series u(x) =

∞

an xn ,

(14.74)

n=0

into both sides of equation (14.73) leads to ⎛ ⎛ ∞ ∞ 2 ⎞⎞ x ⎝T2 (ex−t ) ⎝1 − an xn = T1 (1 − ex + e2x ) + an tn ⎠⎠ dt, n=0

0

n=0

(14.75) where T1 and T2 are the Taylor series about x = 0 and about t = 0 respectively. Evaluating the integral at the right side, using a0 = 1, we ﬁnd a1 + 2a2 x + 3a3 x2 + 4a4 x3 + 5a5 x4 + · · · 3 1 7 1 2 2 − a1 x2 + − a1 − a2 − a1 x3 =1+x+ 2 6 3 3 3 1 2 1 5 1 1 1 − a − a2 − a1 − a3 − a1 a2 x4 + 8 12 1 6 12 2 2 1 31 1 1 1 2 2 1 1 − a1 − a21 − a2 − a22 − a4 − a1 a3 − a3 − a1 a2 x5 + 120 60 60 30 5 5 5 10 10 +O(x6 ). (14.76)

438

14 Nonlinear Volterra Integro-Diﬀerential Equations

Equating the coeﬃcients of like powers of x in both sides, and solving the system of equations we obtain 1 an = , n 0. (14.77) n! This gives the exact solution by (14.78) u(x) = ex . Example 14.10 Solve the nonlinear Volterra integro-diﬀerential equation by using the series solution method x 5 1 cos(x − t)(1 + u2 (t))dt, u(0) = 0. u (x) = cos x − sin x + sin(2x) + 3 3 0 (14.79) Substituting u(x) by the series ∞ u(x) = an xn , (14.80) n=0

into both sides of equation (14.79) leads to ∞ 5 1 n an x = T1 cos x − sin x + sin(2x) 3 3 n=0 ⎛ ∞ 2 ⎞ x (T2 (cos(x − t)) ⎝1 + an tn ⎠ dt, + 0

(14.81)

n=0

where T1 and T2 are the Taylor series about x = 0 and about t = 0 respectively. Evaluating the integral at the right side, using a0 = 0, and proceeding as before we ﬁnd (−1)n (14.82) a2n+1 = , a2n = 0, n 0. (2n + 1)! This gives the exact solution by u(x) = sin x.

(14.83)

Example 14.11 Solve the nonlinear Volterra integro-diﬀerential equation by using the series solution method x 1 1 u (x) = x−sin x−cos x− sin(2x)+ (x−t)(1−u2(t))dt, u(0) = u (0) = 1. 2 4 0 (14.84) Substituting u(x) by the series ∞ an xn , (14.85) u(x) = n=0

into both sides of equation (14.84) leads to

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

∞

n

an x

n=0

439

1 1 x − sin x − cos x − sin(2x) 2 4 ⎛ ⎛ 2 ⎞⎞ x ∞ ⎝(x − t) ⎝1 − an tn ⎠⎠ dt, +

= T1

0

(14.86)

n=0

where T1 is the Taylor series about x = 0. Evaluating the integral at the right side, using a0 = 1, a1 = 1, and proceeding as before we ﬁnd (−1)n (−1)n , a2n+1 = , n 0. (14.87) a2n = (2n)! (2n + 1)! Consequently, the series solution is given by ∞ ∞ (−1)n 2n (−1)n 2n+1 x + x u(x) = , (14.88) (2n)! (2n + 1)! n=0 n=0 that converges to the exact solution u(x) = cos x + sin x.

(14.89)

Example 14.12 Solve the nonlinear Volterra integro-diﬀerential equation by using the series solution method x 2 2 (1 + u2 (t))dt, (14.90) u (x) = 2 sec x(1 + 3 tan x) − tan x + 0

with initial conditions u(0) = u (0) = 0, u (0) = 1. Substituting u(x) by the series ∞ an xn , (14.91) u(x) = n=0

into both sides of equation (14.90), evaluating the integral at the right side, using a0 = a2 = 0, a1 = 1, and proceeding as before we ﬁnd 1 2 , a1 = 1, a3 = , a 5 = 3 15 (14.92) 17 , a2k = 0, a7 = k 0, 315 Consequently, the series solution is given by 2 17 7 1 x + ··· , (14.93) u(x) = x + x3 + x5 + 3 15 315 that converges to the exact solution u(x) = tan x.

(14.94)

Exercises 14.2.3 Solve the following nonlinear Volterra integro-diﬀerential equations by using the series solution method

440

14 Nonlinear Volterra Integro-Diﬀerential Equations x 1. u (x) = −x + tanh x(1 − sech x) + (1 − u2 (t))dt, u(0) = 1 0

1 3 2. u (x) = − x + cosh x + sinh(2x) + 2 4

x 0

(1 − u2 (t))dt, u(0) = 0

x 1 1 3. u (x) = − x + sin x + cos x + sin(2x) + (x − t)(1 − u2 (t))dt, u(0) = −1 2 4 0 x ex+t u2 (t)dt, u(0) = 1 4. u (x) = 1 − 2 cosh x + 0

5. u (x) = 1 − 2 sinh x +

x

0

ex+t u2 (t)dt, u(0) = 1, u (0) = −1

6. u (x) = 1 + ex (1 − 2x) − e2x + 7. u (x) = 2xex + e2x +

x 0

x

0

ex−t u2 (t)dt, u(0) = 2, u (0) = 1

ex−t (1 − u2 (t))dt, u(0) = 2, u (0) = u (0) = 1

1 1 8. u (x) = − x − 8 cos(2x) − sin(4x) + 2 8 u(0) = u (0) = 0, u (0) = 2

x 0

(1 − u2 (t))dt,

14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind The standard form of the nonlinear Volterra integro-diﬀerential equation [1– 3] of the ﬁrst kind is given by x x K1 (x, t)F (u(t))dt + K2 (x, t)u(i) (t)dt = f (x), (14.95) 0

0

where u(i) (x) is the i th derivative of u(x). For this equation, the kernels K1 (x, t) and K2 (x, t), and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x). For the determination of the exact solution, the initial conditions should be prescribed. The nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind (14.95) can be converted to a nonlinear Volterra integral equation of the second kind by integrating the second integral in (14.95) by parts. The nonlinear Volterra integro-diﬀerential equations of the ﬁrst kind will be handled in this section by the combined Laplace transform-Adomian decomposition method and by converting it to a nonlinear Volterra integro-diﬀerential equation of the second kind.

14.3.1 The Combined Laplace Transform-Adomian Decomposition Method The combined Laplace transform-Adomian decomposition method was used in the previous section for solving nonlinear Volterra integro-diﬀerential equa-

14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind

441

tions of the second kind. The analysis will be focused on equations where the kernels K1 (x, t) and K2 (x, t) of (14.95) are diﬀerence kernels. This means that each kernel depends on the diﬀerence (x − t). Recall that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x f1 (x − t)f2 (t)dt = F1 (s)F2 (s). (14.96) L{(f1 ∗ f2 )(x)} = L 0

Recall that L{u(n) (x)} = sn L{u(x)} − sn−1 u(0) − sn−2 u (0) − · · · u(n−1) (0). Taking Laplace transform of both sides of (14.95) gives L{K1 (x − t) ∗ F (u(x))} + L{K2 (x − t) ∗ u(i) (x)} = L{(f (x)}, so that K1 (s)L{F (u(x))} + K2 (s)L{u(i) (x)} = φ(s),

(14.97) (14.98) (14.99)

where φ(s) = L{f (x)}, K1 (s) = L{K1 (x)}, K2 (s) = L{K2 (x)}. Using (14.97) and solving for U (s) we ﬁnd φ(s) + K2 (s)Γ(s) − K1 (s)L{F (u(x))} U (s) = , si K2 (s) where Γ(s) = si−1 u(0) + si−2 u (0) + · · · + u(i−1) (0), U (s) = L{u(x)}.

(14.100)

(14.101)

(14.102)

The combined Laplace transform-Adomian decomposition method can be used eﬀectively in (14.101) provided that K1 (s) = 0. (14.103) lim s→∞ si K2 (s) To overcome the diﬃculty of the nonlinear term F (u(x)), we apply the Adomian decomposition method for handling (14.101). To achieve this goal, we ﬁrst represent the linear term u(x) at the left side by an inﬁnite series of components given by ∞ u(x) = un (x), (14.104) n=0

where the components un (x), n 0 will be recursively determined. However, the nonlinear term F (u(x)) at the right side of (14.101) will be represented by an inﬁnite series of the Adomian polynomials An in the form ∞ An (x), (14.105) F (u(x)) = n=0

where the Adomian polynomials An , n 0 are given by n 1 dn i An = F λ ui , n = 0, 1, 2, · · · . n! dλn i=0 λ=0

(14.106)

442

14 Nonlinear Volterra Integro-Diﬀerential Equations

Substituting (14.104) and (14.105) into (14.101) leads to + ∞ 1 1 1 1 un (x) = u(0) + 2 u (0) + · · · + i u(i−1) (0) + i L φ(s) s s s s K2 (s) n=0 + ∞ K1 (s) An (x) . L − i s K2 (s) n=0 (14.107) The Adomian decomposition method admits the use of the following recursive relation 1 1 1 1 U0 (s) = u(0) + 2 u (0) + · · · + i u(i−1) (0) + i φ(s), s s s s K2 (s) (14.108) K1 (s) L{uk+1 (x)} = − i L{Ak (x)}, k 0, s K2 (s) provided that K1 (s) = 0. (14.109) lim s→∞ si K2 (s) For example if K2 (x, t) = cosh(x − t), K1 (x, t) = x − t , and the equation includes u (x), then s2 = 1. (14.110) lim 2 s→∞ s − 1 In such a problem, the combined Laplace transform-Adomian decomposition method cannot be used. Instead another approach should be used to handle this case. Applying the inverse Laplace transform to the ﬁrst part of (14.108) gives u0 (x), that will deﬁne A0 . This in turn will lead to the complete determination of the components of uk , k 0. The analysis presented above will be illustrated by the following examples. Example 14.13 Solve the following nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind by the combined Laplace transform-Adomian decomposition method x x 7 1 1 (x − t)u2 (t)dt + (x − t)u (t)dt = + x2 − cos x + cos(2x), u(0) = 0. 8 4 8 0 0 (14.111) Taking Laplace transforms of both sides gives 1 1 s 1 7 s + 3− 2 + , (14.112) L{u2 }(s) + 2 (sU (s) − u(0)) = 2 2 s s 8s 2s s + 1 8(s + 4) where by using the given initial condition and solving for U (s) we obtain 1 s2 1 7 s2 + − L{u2 }(s). U (s) = + 2 − 2 (14.113) 8 2s s + 1 8(s2 + 4) s Substituting the series assumption for U (s) and the Adomian polynomials for u2 (x), and using the recursive relation (14.108) we obtain

14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind

443

1 s2 7 s2 + 2− 2 + , 8 2s s + 1 8(s2 + 4) (14.114) 1 L{uk+1 (x)} = − L{Ak (x)}, k 0. s Taking the inverse Laplace transform of both sides of the ﬁrst part of (14.114), and using the recursive relation (14.114) gives 1 7 5 x + ··· , u0 (x) = x + x3 − 3! 120 1 1 (14.115) u1 (x) = − x3 − x5 + · · · , 3 15 2 5 x + ··· . u2 (x) = 15 The series solution is therefore given by 1 1 (14.116) u(x) = x − x3 + x5 + · · · , 3! 5! that converges to the exact solution u(x) = sin x. (14.117) U0 (s) =

Example 14.14 Solve the following nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind by the combined Laplace transform-Adomian decomposition method x x 1 1 1 (x − t)u2 (t)dt + ex−t u (t)dt = − − x + xex + e2x , u(0) = 1. 4 2 4 0 0 (14.118) Taking Laplace transforms of both sides gives 1 1 1 1 s 1 (sU (s) − u(0)) = − − 2 + , L{u2 }(s) + + s2 s−1 4s 2s (s − 1)2 4(s − 2) (14.119) where by using the given initial condition and solving for U (s) we obtain 1 s−1 s 1 1 1 s−1 U (s) = + + − − 2+ − 3 L{u2 }(s). s s 4s 2s (s − 1)2 4(s − 2) s (14.120) Substituting the series assumption for U (s) and the Adomian polynomials for u2 (x), and using the recursive relation (14.108) we obtain 1 1 1 s−1 s 1 − − 2+ , U0 (s) = + + s s 4s 2s (s − 1)2 4(s − 2) (14.121) s−1 L{uk+1 (x)} = − 3 L{Ak (x)}, k 0. s Taking the inverse Laplace transform of both sides of the ﬁrst part of (14.121), and using the recursive relation (14.121) gives

444

14 Nonlinear Volterra Integro-Diﬀerential Equations

1 1 1 u0 (x) = 1 + x + x2 + x3 + x4 + x5 + · · · , 3 8 24 1 1 1 1 u1 (x) = − x2 − x3 − x4 − x5 + · · · , 2 6 6 12 1 4 1 5 x + x + ··· . u2 (x) = 12 20 The series solution is therefore given by 1 1 1 u(x) = 1 + x + x2 + x3 + x4 + x5 + · · · , 3! 4! 5! that converges to the exact solution u(x) = ex .

(14.122)

(14.123) (14.124)

Example 14.15 Solve the following nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind by the combined Laplace transform-Adomian decomposition method x x 15 1 1 cos2 (2x), (x − t)u2 (t)dt + (x − t)u (t)dt = − + x2 + cos(2x) − 16 4 16 0 0 (14.125) where u(0) = 1, u (0) = 0. Taking Laplace transforms of both sides and using the initial conditions we obtain 15 1 2 1 1 1 cos2 (2x) − 2 L{u2 }(s). (14.126) U (s) = +L − + x +cos(2x) − s 16 4 16 s Proceeding as before leads to 3 1 4 u0 (x) = 1 − x2 + x4 + x6 + · · · , 2 3 45 1 2 1 4 7 6 u1 (x) = − x + x − x + · · · , 2 4 72 (14.127) 1 4 1 6 u2 (x) = x + x + ··· , 12 15 1 6 u3 (x) = − x + · · · , 72 The series solution is therefore given by 1 1 1 (14.128) u(x) = 1 − (2x)2 + (2x)4 − (2x)6 + · · · , 2! 4! 6! that converges to the exact solution u(x) = cos(2x). (14.129) Example 14.16 Solve the following nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind by the combined Laplace transform-Adomian decomposition method x x 1 1 1 2 (x − t)u (t)dt + (x − t)u (t)dt = − − 3x + x2 + 3 sinh x + cosh2 x, 4 4 4 0 0 (14.130)

14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind

445

where u(0) = 1, u (0) = 1. Taking Laplace transforms of both sides and using the initial conditions we obtain 1 1 1 1 1 1 U (s) = + 2 + L{− − 3x + x2 + 3 sinh x + cosh2 x} − 2 L{u2 }(s). s s 4 4 4 s (14.131) Proceeding as before leads to 1 1 1 1 u0 (x) = 1 + x + x2 + x3 + x4 + x5 + · · · , 2 2 12 40 1 1 1 1 (14.132) u1 (x) = − x2 − x3 − x4 − x5 + · · · , 2 3 6 10 1 4 1 u2 (x) = x + x5 + · · · , 12 12 The series solution is therefore given by 1 1 u(x) = 1 + x + x3 + x5 + · · · , (14.133) 3! 5! that converges to the exact solution u(x) = 1 + sinh x.

(14.134)

Exercises 14.3.1 Solve the following nonlinear Volterra integro-diﬀerential equations of the ﬁrst kind by using the combined Laplace transform-Adomian decomposition method x x 1 1 1 1. (x − t)u2 (t)dt + (x − t)u (t)dt = − x + x2 + sin x − cos(2x), u(0) = 1 8 4 8 0 0 x x 1 1 1 2 2 (x − t)u (t)dt + (x − t)u (t)dt = − − x + x + sinh x + cosh(2x), 2. 8 4 8 0 0 u(0) = 1 x (x − t)u2 (t)dt + 3. 0

u(0) = 1 x ex−t u2 (t)dt + 4. 0 x

5. 0 x

6. 0

ex−t u2 (t)dt +

x 0

x

ex−t u (t)dt = −4 + 3ex + 5xex + e2x , u(0) = 3, u (0) = 1

0

1 1 1 x + x2 + sin x − cos x − sin(2x), 2 2 4

ex−t u (t)dt = −1 + 3xex + e2x , u(0) = 2

0 x

(x − t)u2 (t)dt +

(x − t)u (t)dt = 1 −

x 0

(x − t)u (t)dt = −

1 1 − 2x + x2 + sin(2x) 16 4

1 + cos2 x, u(0) = 0, u (0) = 2 16 x x 1 6 (x − t)u2 (t)dt + (x − t)u (t)dt = x2 − 7. x , u(0) = 0, u (0) = 0 36 0 0 x x 10 1 4 8. (x − t)u3 (t)dt + (x − t)u (t)dt = − − x + ex + e3x , 9 3 9 0 0 u(0) = 1, u (0) = 1

446

14 Nonlinear Volterra Integro-Diﬀerential Equations

14.3.2 Conversion to Nonlinear Volterra Equation of the Second Kind In this section we will convert the nonlinear Volterra integro-diﬀerential equation [1–3] of the ﬁrst kind x x K1 (x, t)F (u(t))dt + K2 (x, t)u(n) (t)dt = f (x), K2 (x, x) = 0, 0

0

(14.135) to a nonlinear Volterra integral equation of the second kind or nonlinear Volterra integro-diﬀerential equation of the second kind. Without loss of generality, we will study the cases of the ﬁrst and second order derivatives of the form x

0

x

K1 (x, t)F (u(t))dt +

and x 0

0

K1 (x, t)F (u(t))dt +

x 0

K2 (x, t)u (t)dt = f (x), K2 (x, x) = 0, (14.136)

K2 (x, t)u (t)dt = f (x), K2 (x, x) = 0. (14.137)

However, equations of higher order can be handled in a similar manner. Integrating the second integral in (14.136) by parts gives the nonlinear Volterra integral equation x x ∂K2 (x, t) u(t)dt K1 (x, t)F (u(t))dt + K2 (x, x)u(x) − K2 (x, 0)u(0) − ∂t 0 0 = f (x), (14.138) or equivalently x K2 (x, 0) 1 f (x) ∂(K2 (x, t)) + u(0) + u(t)dt u(x) = K2 (x, x) K2 (x, x) K2 (x, x) 0 ∂t (14.139) x 1 − K1 (x, t)F (u(t))dt, K2 (x, x) = 0. K2 (x, x) 0 Equation (14.139) is the nonlinear Volterra integral equation of the second kind that was handled in this chapter by distinct methods. In a like manner, we integrate the second integral in (14.137) by parts to obtain the nonlinear Volterra integro-diﬀerential equation of the second kind x K1 (x, t)F (u(t))dt + K2 (x, x)u (x) − K2 (x, 0)u (0) 0 (14.140) x ∂K2 (x, t) u (t)dt = f (x), K2 (x, x) = 0. − ∂t 0 or equivalently

14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind

447

x ∂(K2 (x, t)) K2 (x, 0) f (x) 1 + u (0) + u (t)dt K2 (x, x) K2 (x, x) K2 (x, x) 0 ∂t (14.141) x 1 K1 (x, t)F (u(t))dt, K2 (x, x) = 0. − K2 (x, x) 0 It is important to notice that if the nonlinear Volterra integral equation of the ﬁrst kind contains the ﬁrst derivative of u(x), then the conversion process will give a nonlinear Volterra integral equation of the second kind as shown by (14.139). However, if the nonlinear Volterra integral equation of the ﬁrst kind contains u(i) (x), i 2, then integrating the second integral once will give a nonlinear Volterra integro-diﬀerential equation of the second kind as shown by (14.141). Both types of equations were examined before. In what follows, the ﬁrst two illustrative examples will be handled by the modiﬁed decomposition method and the other two examples will be handled by using the variational iteration method. Other methods can be used as well.

u (x) =

Example 14.17 Convert the nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind x x 1 1 1 (x−t)u2 (t)dt+ ex−t u (t)dt = xex + e2x − − x, u(0) = 1, (14.142) 4 4 2 0 0 to a nonlinear Volterra integral equation of the second kind and solve it. Integrating the second integral by parts, using the initial condition, and solving for u(x) we ﬁnd x x 1 1 1 u(x) = ex +xex + e2x − − x− (x−t)u2 (t) dt− ex−tu(t) dt. (14.143) 4 4 2 0 0 We select the modiﬁed decomposition method, therefore we use the recurrence relation approximations u0 (x) = ex + xex , x x 1 2x 1 1 2 u1 (x) = e − − x − (x − t)u0 (t)dt − ex−tu0 (t)dt, (14.144) 4 4 2 0 0 = −xex + · · · . Canceling the noise term xex from u0 (x) gives the exact solution u(x) = ex .

(14.145)

Example 14.18 Convert the nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind x x 1 1 2 u (t) dt+ (x−t+1) u(t) dt = sin x+cos x+ sin(2x)−1− x, u(0) = 1, 4 2 0 0 (14.146) to a nonlinear Volterra integral equation of the second kind and solve it. Integrating the second integral by parts, using the initial condition, and solving for u(x) we ﬁnd

448

14 Nonlinear Volterra Integro-Diﬀerential Equations

x x 1 1 2 u(x) = sin x + cos x + sin(2x) + x − u (t) dt − u(t) dt. (14.147) 4 2 0 0 We again select the modiﬁed decomposition method, therefore we use the recurrence relation approximations u0 (x) = sin x + cos x, x x 1 1 2 u1 (x) = sin(2x) + x − u0 (t)dt − u0 (t)dt, (14.148) 4 2 0 0 = − sin x + · · · . Canceling the noise term sin x from u0 (x) gives the exact solution u(x) = cos x. (14.149) Example 14.19 Convert the nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind x x 1 1 u2 (t)dt+ (x−t+1)u (t)dt = sin x−cos x− sin(2x)+1+ x, (14.150) 4 2 0 0 with u(0) = u (0) = 1, to a nonlinear Volterra integro-diﬀerential equation of the second kind and solve it. Integrating the second integral by parts, using the initial conditions, and solving for u(x) we ﬁnd x x 3 1 u2 (t)dt− u (t)dt. (14.151) u (x) = 2 + x+ sin x− cos x− sin(2x)− 2 4 0 0 This equation will be solved by using the variational iteration method. The correction functional for this equation is given by un+1 (x) = un (x) x t 1 3 un (t) − 2 − t − sin t + cos t + sin(2t) + − (un (r) + u2n (r))dr dt. 2 4 0 0 (14.152) We can use the initial conditions to select u0 (x) = 1+x. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1 + x, 1 1 1 u1 (x) = 1 + x − x3 − x4 − x5 + · · · , 3! 4! 5! (14.153) 1 1 u2 (x) = 1 + x − x3 + x5 + · · · , 3! 60 1 1 u3 (x) = 1 + x − x3 + x5 + · · · , 3! 5! and so on. The VIM gives the exact solution by u(x) = 1 + sin x.

(14.154)

Example 14.20 Convert the nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind

14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind

x

449

x

3 1 ex−t u (t) dt = − + ex (2 − x) − e2x , (14.155) 2 2 0 0 with u(0) = 1, u (0) = 2, to a nonlinear Volterra integro-diﬀerential equation of the second kind and solve it. Integrating the second integral by parts, using the initial equation, and solving for u(x) we ﬁnd x x 3 1 (t2 −u2 (t)) dt− ex−t u (t) dt. (14.156) u (x) = − +ex (4−x)− e2x − 2 2 0 0 This equation will be solved by using the variational iteration method. The correction functional for this equation is given by un+1 (x) = un (x) t x 3 1 (r2 − u2n (r) + et−r un (r))dr dt. − un(t) + − et (4 − t) + e2t + 2 2 0 0 (14.157) Using the initial conditions to select u0 (x) = u(0) + xu (0) = 1 + 2x, we obtain the following successive approximations u0 (x) = 1 + 2x, 1 1 1 u1 (x) = 1 + 2x + x2 + x3 + x4 + · · · , 2! 3! 4! 1 2 1 3 1 u2 (x) = 1 + 2x + x + x + x5 + · · · , (14.158) 2! 3! 60 1 2 1 3 1 4 u3 (x) = 1 + 2x + x + x + x + · · · , 2! 3! 4! 1 2 1 3 1 1 u4 (x) = 1 + 2x + x + x + x4 + x5 + · · · , 2! 3! 4! 5! and so on. The VIM gives the exact solution by (t2 − u2 (t)) dt +

u(x) = x + ex .

(14.159)

Exercises 14.3.2 In Exercises 1–4, convert nonlinear Volterra integro-diﬀerential equations of the ﬁrst kind to nonlinear Volterra integral equations of the second kind and solve the resulting equations by any method x x 1 (cosh2 t − u2 (t))dt + cosh(x − t)u (t)dt = −x − 2 sinh x + x sinh x, u(0) = 2 1. 2 0 0 x x 1 (cos2 t − u2 (t))dt + cos(x − t)u (t)dt = −x − 2 sin x − x sin x, u(0) = 2 2. 2 0 0 x x 5 1 (t2 − u2 (t))dt + ex−t u (t)dt = − + ex (3 − x) − e2x , u(0) = 1 3. 2 2 0 0 x x 1 1 2 (1 − u (t))dt + cos(x − t)u (t)dt = sin x(2 sin x + x + 1) + x cos x, 4. 2 2 0 0 u(0) = −1

450

14 Nonlinear Volterra Integro-Diﬀerential Equations

In Exercises 5–8, convert nonlinear Volterra integro-diﬀerential equations of the ﬁrst kind to a nonlinear Volterra integro-diﬀerential equations of the second kind and solve the resulting equations by any method x x 1 1 5. (u2 (t) − 1)dt + cos(x − t)u (t)dt = sin x(2 sin x − x − 1) − x cos x, 2 2 0 0 u(0) = 1, u (0) = 1 x (sin2 t − u2 (t))dt + 6. 0

x 0

1 1 cos(x − t)u (t)dt = − x3 − (x + 4) sin x 3 2

+2x cos x, u(0) = 0, u (0) = 2 x x ex−4t u2 (t)dt + ex−2t u (t)dt = 5xex , u(0) = 1, u (0) = 2 7.

0 x

8. 0

0

2

(x − t)u (t)dt +

x 0

1 cosh(x − t)u (t)dt = − (1 + x2 ) 4

1 1 + x sinh x + cosh2 x, u(0) = 0, u (0) = 1 2 4

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations In this section, we will study systems of nonlinear Volterra integro-diﬀerential equations of the second kind given by x % 1 (x, t)F%1 (v(t)) dt, u(i) (x) = f1 (x) + K1 (x, t)F1 (u(t)) + K 0 (14.160) x (i) % % K2 (x, t)F2 (u(t)) + K2 (x, t)F2 (v(t)) dt. v (x) = f2 (x) + 0

The nonlinear functions Fi , F%i , i = 1, 2 of the unknown functions u(x), v(x) occur inside the integral sign whereas the derivatives of u(x), v(x) appear % i (x, t), and the mostly outside the integral sign. The kernels Ki (x, t) and K functions fi (x), i = 1, 2 are given real-valued functions. To determine the exact solutions for the system (14.160), the initial conditions u(j−1) (0) and v (j−1) (0), 1 j i should be prescribed. There is a variety of numerical and analytical methods that are usually used for solving the systems of nonlinear Volterra integro-diﬀerential equations (14.160). However, in this section, we will concern ourselves with two methods, namely the variational iteration method and the combined Laplace transform-Adomian method.

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

451

14.4.1 The Variational Iteration Method The variational iteration method (VIM) was used to handle Volterra integral equations and Volterra integro-diﬀerential equations. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the so called Adomian polynomials that we need to express nonlinear terms. The correction functionals for the Volterra system of integro-diﬀerential equations (14.160) are given by x t un+1 (x) = un (x) + λ(t) u(i) (t) − f (t) − γ (t, r)dr dt, 1 1 n 0 0 (14.161) x t vn+1 (x) = vn (x) + where

0

λ(t) vn(i) (t) − f2 (t) −

0

γ2 (t, r)dr dt.

% 1 (t, r)F%1 (˜ γ1 (t, r) = K1 (t, r)F1 (˜ un (r)) + K vn (r)), % 2 (t, r)F%2 (˜ γ2 (t, r) = K2 (t, r)F2 (˜ un (r)) + K vn (r)).

(14.162)

The variational iteration method is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ that can be identiﬁed optimally via integration by parts and by using a restricted variation. Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), n 0 and vn+1 (x), n 0 of the solutions u(x) and v(x). The zeroth approximations u0 (x) and v0 (x) can be any selective functions. However, using the initial conditions are preferably used for the selective zeroth approximations u0 and v0 (x) as will be seen later. Consequently, the solutions are given by (14.163) u(x) = lim un (x), v(x) = lim vn (x). n→∞

n→∞

The VIM will be illustrated by studying the following systems of nonlinear Volterra integro-diﬀerential equations of the second kind. Example 14.21 Use the VIM to solve the system of nonlinear Volterra integro-diﬀerential equations x 1 2 1 4 u (x) = 1 − x + x − x + ((x − t)u2 (t) + v 2 (t))dt, 2 12 0 (14.164) x 3 2 1 4 2 2 v (x) = −1 − x − x − x + (u (t) + (x − t)v (t))dt, 2 12 0 where u(0) = 1, v(0) = 1. The correction functionals for this system are

452

14 Nonlinear Volterra Integro-Diﬀerential Equations

x 1 2 1 4 un+1 (x) = un (x) − un (t) − 1 + t − t + t − I1 (t) dt, 2 12 0 (14.165) x 3 1 vn (t) + 1 + t + t2 + t4 − I2 (t) dt, vn+1 (x) = vn (x) − 2 12 0 where t

I1 (t) =

0

I2 (t) =

0

t

((t − r)u2n (r) + vn2 (r))dr,

(14.166) (u2n (r)

+ (t −

r)vn2 (r))dr,

and λ = −1 for ﬁrst order integro-diﬀerential equation. Selecting u0 (x) = u(0) = 1 and v0 (x) = v(0) = 1 gives the successive approximations u0 (x) = 1,

v0 (x) = 1, 1 1 1 u1 (x) = 1 + x + x3 − x4 − x5 , 3 6 60 1 1 v1 (x) = 1 − x − x3 − x5 , 3 60 1 3 1 3 1 4 1 4 1 u2 (x) = 1 + x + x − x + x − x − x5 + · · · , 3 3 6 6 30 1 1 4 1 4 1 3 1 3 v2 (x) = 1 − x + x − x + x − x + x5 + · · · , 3 3 6 6 30 1 5 1 5 u3 (x) = 1 + x + x − x + ··· , 30 30 1 5 1 5 x − x + ··· , v3 (x) = 1 − x + 30 30 and so on. It is obvious that the noise terms appear in each approximation, hence the exact solutions are given by (u(x), v(x)) = (1 + x, 1 − x). (14.167) Example 14.22 Use the VIM to solve the system of Volterra integro-diﬀerential equations x 1 1 3 u (x) = ex − e2x − x4 + x + + ((x − t)u2 (t) + (x − t)v 2 (t))dt, 2 6 2 0 x ((x − t)u2 (t) − (x − t)v 2 (t))dt, v (x) = 7ex − 4xex − 4x − 7 + 0

(14.168) where u(0) = 1, v(0) = −1. The correction functionals for this system are given by

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

453

1 2t 1 4 3 t un+1 (x) = un (x) − un(t) − e + e + t − t − − I1 (t) dt, 2 6 2 0 x

vn+1 (x) = vn (x) − vn (t) − 7et + 4tet + 4t + 7 − I2 (t) dt, x

0

where

(14.169)

I1 (t) = I2 (t) =

t 0

((t − r)u2n (r) + (t − r)vn2 (r))dr, (14.170)

t 0

((t −

r)u2n (r)

− (t −

r)vn2 (r))dr,

and λ = −1 for ﬁrst-order integro-diﬀerential equations. We can use the initial conditions to select u0 (x) = u(0) = 1 and v0 (x) = v(0) = −1. Using this selection into the correction functionals gives the following successive approximations v0 (x) = −1, 1 1 1 11 5 x + ··· , u1 (x) = 1 + 2x + x2 + x3 − x4 − 2! 3! 8 120 1 1 5 3 v1 (x) = −1 − x2 − x3 − x4 − x5 + · · · , 2! 3! 24 40 1 1 1 1 u2 (x) = 1 + 2x + x2 + x3 + x4 + x5 + · · · , 2! 3! 4! 5! 1 1 1 1 v2 (x) = −(1 + x2 + x3 + x4 + x5 + · · · ), 2! 3! 4! 5! and so on. The exact solutions are therefore given by (u(x), v(x)) = (x + ex , x − ex ). u0 (x) = 1,

(14.171)

Example 14.23 Use the VIM to solve the system of nonlinear Volterra integro-diﬀerential equations 1 1 1 u (x) = cosh x − sinh2 x − x4 − x2 2 6 2 x + ((x − t)u2 (t) + (x − t)v 2 (t))dt, u(0) = 1, u (0) = 1, 0

v (x) = −(1 + 4x) cosh x + 8 sinh x − 4x x ((x − t)u2 (t) − (x − t)v 2 (t))dt, v(0) = −1, v (0) = 1. + 0

The correction functionals for this system are given by

(14.172)

454

14 Nonlinear Volterra Integro-Diﬀerential Equations

un+1 (x) = un (x) x t 1 1 1 (t − x) un (t) − cosh t + sinh2 t + t4 + t2 − + I1 (t, r)dr dt, 2 6 2 0 0 (14.173) vn+1 (x) = vn (x) t x I2 (t, r)dr dt, (t − x) vn (t) + (1 + 4t) cosh t − 8 sinh t + 4t − + 0

0

where

I1 (t, r) = (t − r)u2 (r) + (t − r)v2 (r), I2 (t, r) = (t − r)u2 (r) − (t − r)v2 (r).

(14.174)

and λ = t − x for second-order integro-diﬀerential equation. Using u0 (x) = 1+x and v0 (x) = −1+x gives the successive approximations v0 (x) = −1 + x, 1 1 1 6 x + ··· , 1 + x + x2 + x4 − 2! 4! 240 1 1 1 6 x + ··· , −1 + x − x2 − x4 − 2! 4! 720 1 1 1 x + 1 + x2 + x4 + x6 + · · · , 2! 4! 6! 1 1 1 x − 1 + x2 + x4 + x6 + · · · , 2! 4! 6!

u0 (x) = 1 + x, u1 (x) = v1 (x) = u2 (x) = v2 (x) =

and so on. The exact solutions are therefore given by (u(x), v(x)) = (x + cosh x, x − cosh x).

(14.175)

Example 14.24 Use the variational iteration method to solve the following system e4x 3 e2x (x − 1) + (3x − 1) + (x + 1) u (x) = ex + 2 4 4 x ((x − 2t)u2 (t) + (x − 4t)v 2 (t))dt, u(0) = 1, u (0) = 1, + 0

e6x 5 e4x (3x − 1) + (5x − 1) + (x + 1) v (x) = 4e2x + 4 6 12 x + ((x − 4t)v 2 (t) + (x − 6t)w2 (t))dt, v(0) = 1, v (0) = 2, 0

e6x 2 e2x (x − 1) + (5x − 1) + (x + 1) w (x) = 9e + 2 6 3 x 2 2 ((x − 6t)w (t) + (x − 2t)u (t))dt, w(0) = 1, w (0) = 3. +

3x

0

(14.176) We select the zeroth approximations as u0 (x) = 1 + x, v0 (x) = 1 + 2x and w0 (x) = 1 + 3x. Proceeding as before, we obtain

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

u0 (x) = 1 + x,

v0 (x) = 1 + 2x,

u1 (x) = 1 + x +

1 2 1 1 1 x + x3 + x 4 + x5 + · · · , 2! 3! 4! 5!

v1 (x) = 1 + 2x +

455

w0 (x) = 1 + 3x,

1 1 1 1 (2x)2 + (2x)3 + (2x)4 + (2x)5 + · · · , 2! 3! 4! 5!

1 1 1 1 (3x)2 + (3x)3 + (3x)4 + (3x)5 + · · · , 2! 3! 4! 5! The exact solutions are therefore given by (u(x), v(x), w(x)) = (ex , e2x , e3x ). (14.177) w1 (x) = 1 + 3x +

Exercises 14.4.1 Use the variational iteration method to solve the following systems of nonlinear Volterra integro-diﬀerential equations x ⎧ 1 4 2 6 ⎪ ⎪ (x) = 2x + + + (x − 2t)(u2 (t) + v(t))dt x x u ⎪ ⎪ 6 15 ⎪ 0 ⎪ ⎨ x 1 4 2 6 1. v x x (x) = −2x − + + (x − 2t)(u(t) + v 2 (t))dt ⎪ ⎪ ⎪ 6 15 0 ⎪ ⎪ ⎪ ⎩ u(0) = 1, v(0) = 1 x ⎧ 1 3 8 ⎪ ⎪ x + (x − 2t)(u2 (t) + v 2 (t)) dt u (x) = 1 + 3x2 + x4 + ⎪ ⎪ 3 14 ⎪ 0 ⎪ ⎨ x 8 6 2. 2 x (x) = 1 − 3x + + (x − 2t)(u2 (t) − v 2 (t))dt v ⎪ ⎪ ⎪ 15 0 ⎪ ⎪ ⎪ ⎩ u(0) = 0, v(0) = 0 x ⎧ ⎪ x 2x ⎪ (x) = e − 2e + 2 + ex−t (u2 (t) + v 2 (t))dt u ⎪ ⎪ ⎪ 0 ⎪ ⎨ x 3. x x (x) = −e − 4xe + ex−t (u2 (t) − v 2 (t))dt v ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ u(0) = 2 v(0) = 0 x ⎧ 7 2 ⎪ ⎪ u (x) = 1 + sin x − sin(2x) − 4x + cos(x − t)(u2 (t) + v 2 (t))dt ⎪ ⎪ 3 3 ⎪ 0 ⎪ ⎨ x 4. sin(x − t)(u2 (t) − v 2 (t))dt v (x) = 1 + (2 − x2 ) sin x − x cos x + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ u(0) = 1, v(0) = −1 x ⎧ 1 ⎪ ⎪ u (x) = 2 − 2x2 − 2x3 − x4 + (x − t + 1)(u2 (t) − v 2 (t))dt ⎪ ⎪ 3 ⎪ 0 ⎪ ⎨ x 2 2 5. v (x) = −2 + x3 + x4 + (x − 2t)(u2 (t) − v 2 (t))dt ⎪ ⎪ ⎪ 3 3 0 ⎪ ⎪ ⎪ ⎩ u(0) = u (0) = 1, v(0) = 1, v (0) = −1

456 14 Nonlinear Volterra Integro-Diﬀerential Equations x ⎧ 5 2 ⎪ ⎪ (x) = cos(x − t)(u2 (t) + v 2 (t))dt sin x + sin(2x) − 4x + u ⎪ ⎪ 3 3 ⎪ 0 ⎪ ⎨ x 6. 2 (x) = (1 − x) sin x + x cos x + sin(x − t)(u2 (t) − v 2 (t))dt v ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ u(0) = 0, u (0) = 2, v(0) = v (0) = 0 x ⎧ ⎪ (x) = (1 − 2x)ex + ⎪ ex−2t (u2 (t) + v(t))dt u ⎪ ⎪ ⎪ 0 ⎪ ⎪ x ⎪ ⎪

x−4t 2 ⎪ ⎪ v (x) = 4e2x − 2xex + ⎨ v (t) + ex−3t w(t) dt e 0 7. ⎪ x ⎪

x−6t 2 ⎪ ⎪ 3x x ⎪ w (t) + ex−t u(t) dt w (x) = 9e − 2xe + e ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ u(0) = u (0) = 1, v(0) = 1, v (0) = 2, w(0) = 1, w (0) = 3 ⎧ 2x e 3 ⎪ ⎪ u (x) = ex + (x − 1) + e4x (3x − 1) + (x + 1) ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ x ⎪ ⎪

⎪ 2 2 ⎪ (x − 2t)u (t) + (x − 4t)v (t) dt + ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ 6x ⎪ ⎪ v (x) = 8e2x + e4x (3x − 1) + 3e (5x − 1) + 5 (x + 1) ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎨ x

8. (x − 4t)v 2 (t) + (x − 6t)w 2 (t) dt + ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ e2x ⎪ 3e6x ⎪ ⎪ w (x) = 27e3x + (x − 1) + (5x − 1) + 2(x + 1) ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ x ⎪ ⎪

⎪ ⎪ ⎪ (x − 6t)w 2 (t) + (x − 2t)u2 (t) dt + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ u(0) = u (0) = 1, v(0) = 1, v (0) = 4, w(0) = 1, w (0) = 9

14.4.2 The Combined Laplace Transform-Adomian Decomposition Method The combined Laplace transform-Adomian decomposition method was used in this chapter to handle nonlinear Volterra integral equations where it worked eﬀectively. We will use the combined Laplace transform-Adomian decomposition method to study systems of nonlinear Volterra integro-diﬀerential equations of the second kind x % 1 (x, t)F%1 (v(t)))dt, u(i) (x) = f1 (x) + (K1 (x, t)F1 (u(t)) + K 0 (14.178) x % 2 (x, t)F%2 (v(t)))dt. (K2 (x, t)F2 (u(t)) + K v (i) (x) = f2 (x) + 0

The nonlinear functions Fi , F%i , i = 1, 2 of the unknown functions u(x), v(x) occur inside the integral sign whereas the derivatives of u(x) and v(x) appear

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

457

% i (x, t), and the mostly outside the integral sign. The kernels Ki (x, t) and K functions fi (x), i = 1, 2 are given real-valued functions. % i (x, t) as diﬀerence kernels We will consider the kernels Ki (x, t) and K where each kernel depends on the diﬀerence x − t. To determine the exact solutions for the system (14.178), the initial conditions u(j−1) (0) and v (j−1) (0), 1 j i should be prescribed. To use the combined-Laplace transform-Adomian decomposition method, recall that the Laplace transforms of the derivatives of u(x) are given by L{u(i) (x)} = si L{u(x)} − si−1 u(0) − si−2 u (0) − · · · − u(i−1) (0). (14.179) Applying the Laplace transforms to both sides of (14.178) gives si L{u(x)} − si−1 u(0) − si−2 u (0) − · · · − u(i−1) (0) % 1 (x − t) ∗ F%1 (v(t))}, = L{f1 (x)} + L{K1 (x − t) ∗ F1 (u(t)) + K si L{v(x)} − si−1 v(0) − si−2 v (0) − · · · − v(i−1) (0) % 2 (x − t) ∗ F%2 (v(t))}, = L{f2 (x)} + L{K2 (x − t) ∗ F2 (u(t)) + K

(14.180)

or equivalently 1 1 1 1 L{u(x)} = u(0) + 2 u (0) + · · · + i u(i−1) (0) + i L{f1 (x)} s s s s 1 % 1 (x − t) ∗ F%1 (v(t))}, + i L{K1 (x − t) ∗ F1 (u(t)) + K s (14.181) 1 1 1 1 L{v(x)} = v(0) + 2 v (0) + · · · + i v (i−1) (0) + i L{f2 (x)} s s s s 1 % 2 (x − t) ∗ F%2 (v(t))}. + i L{K2 (x − t) ∗ F2 (u(t)) + K s To overcome the diﬃculty of the nonlinear terms Fi (u(x)), i = 1, 2, we apply the Adomian decomposition method for handling (14.181). To achieve this goal, we ﬁrst represent the linear terms u(x) and v(x) at the left side by an inﬁnite series of components given by ∞ ∞ u(x) = un (x), v(x) = vn (x), (14.182) n=0

n=0

and the nonlinear terms Fi (u(x)) at the right side of (14.181) by n ∞ 1 dn i F F (u(x)) = An (x), An = λ ui , n = 0, 1, 2, · · · , n! dλn n=0 i=0 λ=0 (14.183) where the Adomian polynomials An , n 0 can be obtained for all forms of nonlinearity. Substituting (14.182) and (14.183) into (14.181) leads to + ∞ 1 1 1 1 un (x) = u(0) + 2 u (0) + · · · + i u(i−1) (0) + i L{f1 (x)} L s s s s n=0 ∞ + 1 + i L{K1 (x − t)}L An (x) s n=0

458

14 Nonlinear Volterra Integro-Diﬀerential Equations

∞

∞ + 1 % 1 (x − t)}L %n (x) , + i L{K A s n=0

+

1 1 1 1 v(0) + 2 v (0) + · · · + i v(i−1) (0) + i L{f2 (x)} s s s s n=0 ∞ + 1 + i L{K2 (x − t)}L Bn (x) s n=0 ∞ + 1 % ˜ + i L{K2 (x − t)}L Bn (x) . s n=0 (14.184) The Adomian decomposition method admits the use of the following recursive relations 1 1 1 1 L{u0 (x)} = u(0) + 2 u (0) + · · · + i u(i−1) (0) + i L{f1 (x)}, s s s s 1 1 % 1 (x − t)}L{A %k (x)}, L{uk+1 (x)} = i L{K1 (x − t)}L{Ak (x)} + i L{K s s and 1 1 1 1 L{v0 (x)} = v(0) + 2 v (0) + · · · + i v (i−1) (0) + i L{f2 (x)}, s s s s 1 1 % 2 (x − t)}L{B %k (x)}, L{vk+1 (x)} = i L{K2 (x − t)}L{Bk (x)} + i L{K s s (14.185) for k 0. The necessary conditions presented in Chapter 1 for Laplace transform method concerning the limit as s → ∞, should be satisﬁed here for a successful use of this method. Applying the inverse Laplace transform to the ˜0 . ﬁrst part of (14.185) gives u0 (x) and v0 (x), that will deﬁne A0 , A˜0 , B0 , B This in turn will lead to the complete determination of the components of uk+1 , vk+1 (x, k 0 upon using the second part of (14.185). The combined Laplace transform Adomian-decomposition method for solving systems of nonlinear Volterra integro-diﬀerential equations of the second kind will be illustrated by studying the following examples. L

vn (x)

=

Example 14.25 Solve the system of nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method x

ex−t u2 (t) + v 2 (t) dt, u(0) = 2, u (x) = ex − 2e2x + 2 + 0 (14.186) x 2

x−t 2 x x u (t) − v (t) dt, v(0) = 0. v (x) = −e − 4xe + 2 + e 0

Notice that the four kernels are K(x − t) = ex−t . Taking Laplace transforms of both sides of (14.186) gives

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

459

L{u (x)} = L{ex − 2e2x + 2} + L{ex−t ∗ (u2 (x) + v2 (x))}, (14.187) L{v (x)} = L{−ex − 4xex } + L{ex−t ∗ (u2 (x) − v2 (x))}, so that 1 2 2 1 sU (s) − u(0) = − + + L{u2 (x) + v 2 (x)}, s−1 s−2 s s−1 (14.188) 1 1 4 2 2 + (x) − v (x)}, sV (s) − v(0) = − − L{u s − 1 (s − 1)2 s−1 or equivalently 1 2 2 2 1 − + L{u2 (x) + v 2 (x)}, U (s) = + + s s(s − 1) s(s − 2) s2 s(s − 1) 4 1 1 − L{u2 (x) − v2 (x)}. + V (s) = − s(s − 1) s(s − 1)2 s(s − 1) (14.189) Substituting the series assumption for U (s) and V (s), and the Adomian polynomials for u2 (x) and v2 (x) as given above in (14.182) and (14.183) respectively, and using the recursive relation (14.185) we obtain 1 2 2 2 + 2− , U0 (s) = + s s(s − 1) s s(s − 2) 1 Uk+1 (s) = L{Ak (x) + Bk (x)}, s(s − 1) and 1 4 , V0 (s) = − − s(s − 1) s(s − 1)2 (14.190) 1 Vk+1 (s) = L{Ak (x) − Bk (x)}. s(s − 1) Recall that the Adomian polynomials for u2 (x) and v 2 (x) are given by A0 (x) = u20 , B0 (x) = v02 , A1 (x) = 2u0 u1 , B1 (x) = 2v0 v1 , (14.191) B2 (x) = 2v0 v2 + v12 , A2 (x) = 2u0 u2 + u21 , A3 (x) = 2u0 u3 + 2u1 u2 , B3 (x) = 2v0 v3 + 2v1 v2 . Taking the inverse Laplace transform of both sides of (14.190), and using the recursive relation (14.190) gives 3 7 5 31 5 x + ··· , u0 (x) = 2 + x − x2 − x3 − x4 − 2 6 8 120 4 2 (14.192) u1 (x) = 2x2 + x3 − x5 + · · · , 3 15 2 2 u2 (x) = x4 + x5 + · · · , 3 5 and

460

14 Nonlinear Volterra Integro-Diﬀerential Equations

5 3 13 17 5 x + ··· , v0 (x) = −x − x2 − x3 − x4 − 2 2 24 120 4 1 2 v1 (x) = 2x2 + x3 − x4 − x5 , · · · , 3 6 3 2 4 4 5 v2 (x) = x + x + · · · . 3 5 Using (14.182) gives the series solutions by 1 3 1 4 1 5 1 2 u(x) = 1 + 1 + x + x + x + x + x + · · · , 2! 3! 4! 5! 1 2 1 3 1 4 1 v(x) = 1 − 1 + x + x + x + x + x5 + · · · . 2! 3! 4! 5! Consequently, the exact solutions are given by (u(x), v(x)) = (1 + ex , 1 − ex ).

(14.193)

(14.194)

(14.195)

Example 14.26 Solve the system of nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method x 7 2 cos(x − t)(u2 (t) + v 2 (t))dt, u (x) = 1 + sin x − sin(2x) − 4x + 3 3 0 x v (x) = 1 − 7 sin x + 4x cos x + 4x + (x − t)(u2 (t) − v 2 (t))dt, 0

u(0) = 1, v(0) = −1. (14.196) Taking Laplace transforms of both sides of (14.196) gives 7 2 L{u (x)} = L 1 + sin x − sin(2x) − 4x 3 3 +L{cos(x − t) ∗ (u2 (x) + v2 (x))}, (14.197) L{v (x)} = L{1 − 7 sin x + 4x cos x + 4x} +L{(x − t) ∗ (u2 (x) − v2 (x))}, so that 7 4 4 1 s − − L{u2 (x) + v 2 (x)}, sU (s) − u(0) = + + 2 s 3(s2 + 1) 3(s2 + 4) s2 s +1 7 4(s2 − 1) 1 4 1 + 2 sV (s) − v(0) = − 2 + 2 + 2 L{u2 (x) − v 2 (x)}, s s + 1 (s + 1)2 s s (14.198) or equivalently 1 4 4 7 1 1 − − L{u2 (x) + v 2 (x)}, + 2 U (s) = + 2 + s s 3s(s2 + 1) 3s(s2 + 4) s3 s +1 1 4s(s2 − 1) 7 4 1 1 + + 3 + 3 L{u2 (x) − v 2 (x)}. V (s) = − + 2 − s s s(s2 + 1) (s2 + 1)2 s s (14.199)

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

461

Substituting the series assumption for U (s) and V (s), and the Adomian polynomials for u2 (x) and v2 (x) as given above in (14.182) and (14.183) respectively, and using the recursive relation (14.185) we obtain 1 4 4 1 7 U0 (s) = + 2 + − − , s s 3s(s2 + 1) 3s(s2 + 4) s3 1 Uk+1 (s) = 2 L{Ak (x) + Bk }, s +1 and 1 7 4 1 4s(s2 − 1) + 3, V0 (s) = − + 2 − + 2 2 s s s(s + 1) (s + 1)2 s (14.200) 1 Vk+1 (s) = 3 L{Ak (x) − Bk }. s Using the Adomian polynomials, and taking the inverse Laplace transform of both sides of the ﬁrst part of (14.190), and using the recursive relation (14.200) gives 3 1 19 6 x + ··· , u0 (x) = 1 + x − x2 + x4 − 2 8 720 1 1 41 6 x + ··· , u1 (x) = x2 − x4 − x5 + 4 10 360 (14.201) 1 4 1 5 2 6 u2 (x) = x + x − x + · · · , 6 10 15 2 6 x + ··· , u3 (x) = 45 and 1 5 13 6 x + ··· , v0 (x) = −1 + x + x2 − x4 + 2 24 720 1 1 1 (14.202) v1 (x) = x4 − x5 − x6 , · · · , 6 30 30 1 5 1 v2 (x) = x + x6 . 30 60 This in turn gives the series solutions by 1 2 1 4 1 6 u(x) = x + 1 − x + x − x + · · · , 2! 4! 6! (14.203) 1 4 1 1 2 v(x) = x − 1 − x + x − x6 + · · · . 2! 4! 6! Consequently, the exact solutions are given by (u(x), v(x)) = (x + cos x, x − cos x). (14.204) Example 14.27 Solve the system of nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method

462

14 Nonlinear Volterra Integro-Diﬀerential Equations

x

(u2 (t) + v 2 (t))dt, u(0) = 1, v(0) = 1, x v (x) = − cos x − sin x + 2 cos2 x − 2 + (u2 (t) − v 2 (t))dt.

u (x) = cos x − sin x − 2x +

0

0

(14.205) Taking Laplace transforms of both sides of (14.205) gives L{u (x)} = L{cos x − sin x − 2x} + L{1 ∗ (u2 (x) + v2 (x))}, L{v (x)} = L{− cos x − sin x + 2 cos2 x − 2} + L{1 ∗ (u2 (x) − v 2 (x))}, (14.206) so that 1 1 s−1 2 + 2 L{u2 (x) + v 2 (x)}, U (s) = + − s s(s2 + 1) s3 s (14.207) 1 1 s+1 2(s2 + 2) 2 2 2 + L{u (x) − v (x)}. V (s) = − + − s s(s2 + 1) s2 (s2 + 4) s2 s2 Proceeding as in the previous two examples we ﬁnd 3 1 1 1 5 x + ··· , u0 (x) = 1 + x − x2 − x3 + x4 + 2 6 24 120 1 1 u1 (x) = x2 − x4 − x5 + · · · , 6 6 1 4 1 5 u2 (x) = x + x + · · · , 6 6 and 1 1 1 1 v0 (x) = 1 − x − x2 − x3 + x4 + x5 + · · · , 2 2 24 8 2 3 1 4 1 5 (14.208) v1 (x) = x − x − x , · · · , 3 6 6 1 1 v2 (x) = x4 + x5 + · · · . 6 30 Using (14.182) we obtain the series solutions 1 1 1 1 u(x) = 1 − x2 + x4 + · · · + x − x3 + x5 + · · · , 2! 4! 3! 5! (14.209) 1 4 1 3 1 5 1 2 v(x) = 1 − x + x + · · · − x − x + x + · · · . 2! 4! 3! 5! Consequently, the exact solutions are given by (u(x), v(x)) = (cos x + sin x, cos x − sin x). (14.210) Example 14.28 Solve the system of nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

u (x) =

7 x 1 e − e2x − e4x + 3 3

2 1 v (x) = ex + 3e2x + e4x + 3 3

x

ex−t u2 (t) + v 2 (t) dt,

0

0

463

x

ex−t u2 (t) − v2 (t) dt,

(14.211)

u(0) = 1, u (0) = 1, v(0) = 1, v (0) = 2. Taking Laplace transforms of both sides of (14.211) gives 7 x 1 4x 2x + L{ex−t ∗ (u2 (x) + v2 (x))}, e −e − e L{u (x)} = L 3 3 1 4x 2 x 2x + L{ex−t ∗ (u2 (x) − v 2 (x))}, L{v (x)} = L e + 3e + e 3 3 (14.212) so that 1 7 − s2 U (s) − su(0) − u (0) = 3(s − 1) s − 2 1 1 − + L{u2 (x) + v 2 (x)}, 3(s − 4) s − 1 (14.213) 2 3 s2 V (s) − sv(0) − v (0) = + 3(s − 1) s − 2 1 1 + L{u2 (x) − v 2 (x)}, + 3(s − 4) s − 1 or equivalently 1 7 1 1 1 U (s) = 2 + + 2 − 2 − 2 s s 3s (s − 1) s (s − 2) 3s (s − 4) 1 L{u2 (x) + v 2 (x)}, + 2 s (s − 1) (14.214) 2 3 1 1 2 + 2 + 2 V (s) = 2 + + 2 s s 3s (s − 1) s (s − 2) 3s (s − 4) 1 + 2 L{u2 (x) − v 2 (x)}. s (s − 1) Proceeding as before we obtain 1 1 7 9 u0 (x) = 1 + x + x2 − x3 − x4 − x5 + · · · , 2 6 24 40 1 1 7 u1 (x) = x3 + x4 + x5 + · · · , 3 3 30 and

4 3 23 v0 (x) = 1 + 2x + 2x2 + x3 + x4 + x5 + · · · , 3 4 60 1 7 v1 (x) = − x4 − x5 , · · · . 12 60 Using (14.182) gives the series solutions by

(14.215)

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14 Nonlinear Volterra Integro-Diﬀerential Equations

1 2 1 1 1 x + x3 + x4 + x5 + · · · , 2! 3! 4! 5! 1 1 1 1 v(x) = 1 + 2x + (2x)2 + (2x)3 + (2x)4 + (2x)5 + · · · . 2! 3! 4! 5! Consequently, the exact solutions are given by (u(x), v(x)) = (ex , e2x ). u(x) = 1 + x +

(14.216)

(14.217)

Exercises 14.4.2 Use the combined Laplace transform-Adomian decomposition method to solve the following systems of nonlinear Volterra integro-diﬀerential equations by x ⎧ 1 6 ⎪ ⎪ u (x) = 2x − x2 − + (x − t)(u2 (t) + v 2 (t))dt x ⎪ ⎪ ⎪ 15 0 ⎨ x 1 4 1. x v (x) = + (x − t)(u2 (t) − v 2 (t))dt ⎪ ⎪ ⎪ 3 0 ⎪ ⎪ ⎩ u(0) = 1, v(0) = 1 x ⎧ 2 3 1 9 ⎪ 2 ⎪ (x) = 3x − − + (x − t)2 (u2 (t) + v 2 (t))dt x x u ⎪ ⎪ ⎪ 3 126 0 ⎨ x 1 7 2. 2 (x) = −3x − + (x − t)2 (u2 (t) − v 2 (t))dt v x ⎪ ⎪ ⎪ 35 0 ⎪ ⎪ ⎩ u(0) = 1, v(0) = 1 ⎧ x ⎪ ⎪ u (x) = −2 − 3x − 2 sin x + 3 cos x + (u2 (t) + v 2 (t))dt ⎪ ⎪ ⎪ 0 ⎨ x 1 3. (u2 (t) − v 2 (t))dt ⎪ v (x) = −2 + sin x + 2 cos x + sin(2x) + ⎪ ⎪ 2 0 ⎪ ⎪ ⎩ u(0) = 1, v(0) = 2 ⎧ x ⎪ ⎪ u (x) = cos x − 3 sin x + cos(x − t)(u2 (t) + v 2 (t))dt ⎪ ⎪ ⎪ 0 ⎨ x 1 4. sin(2x) − x + (x) = sin x + cos x + (x − t)(u2 (t) − v 2 (t))dt v ⎪ ⎪ ⎪ 2 0 ⎪ ⎪ ⎩ u(0) = 1, v(0) = −1 x ⎧ 8 x 1 1 ⎪ ⎪ ex−t (u2 (t) + v 2 (t))dt e + 2e2x − e4x − e6x + u (x) = ⎪ ⎪ 15 3 5 ⎪ ⎨ 0x 2 x 1 1 5. e + 3e3x − e4x + e6x + ex−t (u2 (t) − v 2 (t))dt ⎪ v (x) = ⎪ ⎪ 15 3 5 0 ⎪ ⎪ ⎩ u(0) = 1, v(0) = 1 x ⎧ ⎪ ⎪ (x) = − sin x − cos x + cos(x + t)(u2 (t) + v 2 (t))dt u ⎪ ⎪ ⎪ 0 ⎨ x 1 6. (u2 (t) − v 2 (t))dt ⎪ v (x) = − sin x − sin(2x) + ⎪ ⎪ 2 0 ⎪ ⎪ ⎩ u(0) = 1, u (0) = 0, v(0) = 0, v (0) = 1

References x ⎧ 3 x 1 −x ⎪ 2x ⎪ (x) = + − e + ex−t (u2 (t) + v(t))dt u e e ⎪ ⎪ ⎪ 2 2 0 ⎨ x 1 1 −2x 7. x −x (x) = − + e + + ex−t (u(t) + v 2 (t))dt (1 + 3x)e e v ⎪ ⎪ ⎪ 3 3 0 ⎪ ⎪ ⎩ u(0) = 1, u (0) = 1, v(0) = 1, v (0) = −1 x ⎧ 2 5 ⎪ 3 ⎪ x u (x) = −2x − 2x − + (u2 (t) + v 2 (t))dt ⎪ ⎪ 5 ⎪ 0 ⎨ x 2 1 8. (x − t)(u2 (t) − v 2 (t))dt v (x) = − x3 − x5 + ⎪ ⎪ ⎪ 3 5 0 ⎪ ⎪ ⎩ u(0) = u (0) = 1, u (0) = 2, v(0) = −v (0) = 1, v (0) = 2

465

References 1. J. Abdelkhani, Higher order methods for solving Volterra integro-diﬀerential equations of the ﬁrst kind, Appl. Math. Comput., 57 (1993) 97–101. 2. P. Linz, A simple approximation method for solving Volterra integro-diﬀerential equations of the ﬁrst kind, J. Inst. Math. Appl., 14 (1974) 211–215. 3. P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, (1985). 4. C. Baker, The Numerical Treatment of Integral Equations, Oxford University Press, London, (1977). 5. H.T. Davis, Introduction to Nonlinear Diﬀerential and Integral Equations, Dover, Publications, New York, (1962). 6. A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 7. R.K. Miller, Nonlinear Volterra Integral Equations, W.A. Benjamin, Menlo Park, CA, (1967). 8. K. Maleknejad and Y. Mahmoudi, Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-diﬀerential equations, Appl. Math. Comput. 145 (2003) 641–653. 9. W.E. Olmstead and R.A. Handelsman, Asymptotic solution to a class of nonlinear Volterra integral equations, II, SIAM J. Appl. Math., 30 (1976) 180–189. 10. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997).

Chapter 15

Nonlinear Fredholm Integral Equations

15.1 Introduction It was stated in Chapter 4 that Fredholm integral equations arise in many scientiﬁc applications. It was also shown that Fredholm integral equations can be derived from boundary value problems. Erik Ivar Fredholm (1866–1927) is best remembered for his work on integral equations and spectral theory. Fredholm was a Swedish mathematician who established the theory of integral equations and his 1903 paper in Acta Mathematica played a major role in the establishment of operator theory. The linear Fredholm integral equations and the linear Fredholm integro-diﬀerential equations were presented in Chapters 4 and 6 respectively. It is our goal in this chapter to study the nonlinear Fredholm integral equations of the second kind and systems of nonlinear Fredholm integral equations of the second kind. The nonlinear Fredholm integral equations of the second kind are characterized by ﬁxed limits of integration of the form b K(x, t)F (u(t))dt. (15.1) u(x) = f (x) + λ a

where the unknown function u(x) occurs inside and outside the integral sign, λ is a parameter, and a and b are constants. For this type of equations, the kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x) such as u2 (x), sin(u(x)), and eu(x) . In this chapter, we will mostly use degenerate or separable kernels. A degenerate or a separable kernel is a function that can be expressed as the sum of product of two functions each depends only on one variable. Such a kernel can be expressed in the form n K(x, t) = gi (x) fi (t). (15.2) i=1

Several analytic and numerical methods have been used to handle the nonlinear Fredholm integral equations. In this text we will use the direct A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

468

15 Nonlinear Fredholm Integral Equations

computation method, the Adomian decomposition method (ADM) combined with the modiﬁed decomposition method (mADM), and the successive substitution method to handle these equations. Systems of nonlinear Fredholm integral equations will be examined as well. For each type of equations we will select the proper methods that facilitate the computational work. The emphasis in this text will be on the use of these methods rather than proving theoretical concepts of convergence and existence. The theorems of existence, uniqueness, and convergence are important and can be found in the literature. The concern will be on the determination of the solutions u(x) of the nonlinear Fredholm integral equations and systems of these equations.

15.2 Existence of the Solution for Nonlinear Fredholm Integral Equations In this section we will present an existence theorem for the solution of nonlinear Fredholm integral equations. The proof of this theorem can be found in [1–2] among other references. The criteria is similar to the criteria presented in Chapter 13 for nonlinear Volterra integral equations. We ﬁrst rewrite the nonlinear Fredholm integral equation of the second kind by b G(x, t, u(t)) dt. (15.3) u(x) = f (x) + λ a

The speciﬁc conditions under which a solution exists for the nonlinear Fredholm integral equation are: (i) The function f (x) is bounded, |f (x)| < R, in a x b. (ii) The function G(x, t, u(t)) is integrable and bounded where |G(x, t, u(t))| < K, in a x, t b. (iii) The function G(x, t, u(t)) satisﬁes the Lipschitz condition |G(x, t, z) − G(x, t, z )| < M |z − z |.

(15.4)

(15.5)

Using the successive approximations method, it is proved in [1] that the series obtained by this method converges uniformly for all values of λ for 1 , (15.6) λ< k(b − a) R where k is the larger of the two numbers K 1 + |λ|K(b−a) and M .

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

469

15.2.1 Bifurcation Points and Singular Points When the nonlinear Fredholm integral equation includes a parameter λ, it is then obvious that the solution of the equation depends on λ. It is possible that λ has a bifurcation point. The bifurcation point is a value of the parameter λ, say λ0 , such that when λ changes through λ0 , then the number of real solutions will be changed. To illustrates this phenomenon, we consider the nonlinear Fredholm integral equation 1 u2 (t) dt. (15.7) u(x) = 3 + λ 0

The solution of this equation is √ (1 − 2λ) ± 1 − 12λ u(x) = . (15.8) 2λ 1 1 The bifurcation point in this problem is λ0 = 12 . For λ 12 , the nonlinear 1 Fredholm equation has two real solutions, but has no real solutions for λ > 12 . 1 This change of λ through the bifurcation point λ0 = 12 caused a change in the number and structure of solutions. However, for λ = 0, we obtain u(x) = 3 by substituting this value of λ in the integral equation itself, whereas u(x) is undeﬁned by substituting λ = 0 into (15.8). Accordingly, the point λ = 0 is called a singular point. More examples will be presented in the next section to address the two phenomena. The following conclusions can be made for nonlinear Fredholm integral equations: (i) The solution of the nonlinear equation may not be unique, there may be more than one solution. (ii) Concerning the bifurcation point, there may be one ore more bifurcation points. This will be seen in the forthcoming examples.

15.3 Nonlinear Fredholm Integral Equations of the Second Kind We begin our study on nonlinear Fredholm integral equations of the second kind of the form b K(x, t)F (u(t))dt, (15.9) u(x) = f (x) + λ a

where the kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x). The unknown function u(x), that will be determined, occurs inside and outside the integral sign. In what follows we will employ four distinct methods, namely the direct computation method, the series solution method, the Adomian decomposition method, and

470

15 Nonlinear Fredholm Integral Equations

the successive approximations method to handle Eq. (15.9). Other methods can be found in this text and in the literature.

15.3.1 The Direct Computation Method In this section, the direct computational method will be applied to solve the nonlinear Fredholm integral equations. The method was used before in Chapters 4 and 6. It approaches nonlinear Fredholm integral equations in a direct manner and gives the solution in an exact form and not in a series form. It is important to point out that this method will be applied for equations where the kernels are degenerate or separable of the form n K(x, t) = gk (x)hk (t). (15.10) k=1

Examples of separable kernels are x − t, xt2 , x3 − t3 , xt4 + x4 t, etc. The direct computation method can be applied as follows: 1. We ﬁrst substitute (15.10) into the nonlinear Fredholm integral equation b K(x, t)F (u(t))dt. (15.11) u(x) = f (x) + λ a

2. This substitution gives u(x) = f (x) b h1 (t) F (u(t))dt + λg2 (x) +λg1 (x) a

b

h2 (t) F (u(t))dt + · · · a

b

+λgn (x)

hn (t) F (u(t))dt. a

(15.12) 3. Each integral at the right side of (15.12) depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (15.12) becomes u(x) = f (x) + λα1 g1 (x) + λα2 g2 (x) + · · · + λαn gn (x), where

(15.13)

b

hi (t) u(t)dt, 1 i n.

αi =

(15.14)

a

4. Substituting (15.13) into (15.14) gives a system of n algebraic equations that can be solved to determine the constants αi , 1 i n. Using the obtained numerical values of αi into (15.13), the solution u(x) of the nonlinear Fredholm integral equation (15.11) follows immediately. It is interesting to point out that we may get more than one value for one or more of αi , 1 i n. This is normal because the equation is nonlinear

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

471

and the solution u(x) is not necessarily unique for nonlinear problems. Linear Fredholm integral equations give unique solutions under the existence conditions presented above. In what follows we present some examples to illustrate the use of this method. Example 15.1 Use the direct computation method to solve the nonlinear Fredholm integral equation 1 u(x) = a + λ u2 (t)dt, a > 0. (15.15) 0

The integral at the right side of (15.15) is equivalent to a constant because it depends only on a function of the variable t with constant limits of integration. Consequently, we rewrite (15.15) as u(x) = a + λα, where

α=

1

u2 (t)dt.

(15.16) (15.17)

0

To determine α, we substitute (15.16) into (15.17) to obtain 1 (a + λα)2 dt, α=

(15.18)

0

where by integrating the right side we ﬁnd λ2 α2 − (1 − 2λa)α + a2 = 0. Solving the quadratic equation (15.19) for α gives √ (1 − 2aλ) ± 1 − 4aλ α= . 2λ2 Substituting (15.20) into (15.16) leads to the exact solutions: √ 1 ± 1 − 4aλ u(x) = . 2λ The following conclusions can be made here:

(15.19)

(15.20)

(15.21)

1. Using λ = 0 into (15.15) gives the exact solution u(x) = a. However, u(x) is undeﬁned by using λ = 0 into (15.21). The point λ = 0 is called the singular point of Equation (15.15). 1 , Equation (15.21) gives only one solution u(x) = 2a. The 2. For λ = 4a 1 point λ = 4a is called the bifurcation point of the equation. This shows that 1 for λ < 4a , then Equation (15.15) gives two real solutions, but has no real 1 solutions for λ > 4a . 1 3. For λ < 4a , Equation (15.21) gives two exact real solutions. Example 15.2 Use the direct computation method to solve the nonlinear Fredholm integral equation

472

15 Nonlinear Fredholm Integral Equations

1

u(x) = x + λ

xtu2 (t)dt.

(15.22)

0

The integral at the right side of (15.22) is equivalent to a constant because it depends only on a function of the variable t with constant limits of integration. Consequently, we rewrite (15.22) as u(x) = (1 + λα)x, where

1

α=

tu2 (t)dt.

(15.23) (15.24)

0

To determine α, we substitute (15.23) into (15.24) to obtain 1 (1 + λα)2 t3 dt, α=

(15.25)

0

where by integrating the right side and solving the resulting equation for α we obtain √ (4 − 2λ) ± 4 1 − λ α= . (15.26) 2λ2 Substituting (15.26) into (15.23) leads to the exact solutions √ 2±2 1−λ u(x) = x. (15.27) λ We next consider the following three cases: 1. Using λ = 0 into (15.22) gives the exact solution u(x) = x. However, u(x) is undeﬁned by using λ = 0 into (15.27). Hence, λ = 0 is called the singular point of Equation (15.22). 2. For λ = 1, Equation (15.27) gives only one solution u(x) = 2x. Therefore, the point λ = 1 is called the bifurcation point of the equation. 3. For λ < 1, Equation (15.27) gives two exact real solutions. This is normal for nonlinear equations. Example 15.3 Use the direct computation method to solve the nonlinear Fredholm integral equation √ 1 3 xu2 (t)dt. (15.28) u(x) = +λ 12 −1 Proceeding as before, we rewrite (15.28) as √ 3 + λαx, u(x) = 12 where 1 α= u2 (t)dt. −1

To determine α, we substitute (15.29) into (15.30) to obtain

(15.29)

(15.30)

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

1

α= −1

473

√ 2 3 + λαt dt, 12

(15.31)

where by integrating the right side and solving the resulting equation for α we obtain √ 6 ± 2 9 − λ2 α= . (15.32) 8λ2 Substituting (15.32) into (15.29) leads to the exact solutions √ √ 3 6 ± 2 9 − λ2 u(x) = + x. (15.33) 12 8λ We consider the following three cases: √

1. Using λ = 0 into (15.28) gives the exact solution u(x) = 123 . However, u(x) is undeﬁned by using λ = 0 into (15.33). Hence, λ = 0 is called the singular point of equation (15.28). √ 2. For λ = ±3, Equation (15.33) gives two solutions u(x) = 123 ± 14 x. Consequently, there are two bifurcation points, namely ±3 for this equation. This shows that for −3 < λ < 3, then equation (15.15) gives two real solutions, but has no real solutions for λ > 3 or λ < −3. 3. For −3 < λ < 3, Equation (15.33) gives two exact real solutions. Example 15.4 Use the direct computation method to solve the nonlinear Fredholm integral equation 9 1 1 2 2 xt u (t)dt. (15.34) u(x) = x + 5 3 −1 Proceeding as before, we rewrite (15.34) as 9 1 u(x) = + α x, 5 3 where 1

α=

t2 u2 (t)dt.

(15.35)

(15.36)

−1

To determine α, we substitute (15.35) into (15.36) to obtain 2 1 9 1 α= (15.37) + α t4 dt, 5 3 −1 where by integrating the right side and solving the resulting equation for α we obtain 18 81 α= , . (15.38) 5 10 Substituting (15.38) into (15.35) leads to the two exact solutions 9 (15.39) u(x) = 3x, x. 2

474

15 Nonlinear Fredholm Integral Equations

Example 15.5 Use the direct computation method to solve the nonlinear Fredholm integral equation 1 5 u(x) = x + x t2 u3 (t)dt. (15.40) 6 0 Proceeding as before, we rewrite (15.40) as 5 + α x, u(x) = 6 where 1

α=

(15.41)

t2 u3 (t)dt.

(15.42)

0

To determine α, we substitute (15.41) into (15.42) to obtain 3 1 5 + α t5 dt, α= (15.43) 6 0 where by integrating the right side and solving the resulting equation for α we obtain √ 21 1 4 . (15.44) α = ,− ± 6 3 2 Substituting (15.44) into (15.41) leads to the three exact solutions √ √ 1 1 u(x) = x, − (1 − 21)x, − (1 + 21)x. (15.45) 2 2 Example 15.6 Use the direct computation method to solve the nonlinear Fredholm integral equation 1 1 11 x + x2 + (xt2 + x2 t)u2 (t)dt. (15.46) u(x) = 35 5 −1 Proceeding as before, we rewrite (15.46) as 11 1 u(x) = +α x+ + β x2 , 35 5 where 1

α= −1

t2 u2 (t)dt,

1

β=

tu2 (t)dt.

(15.47)

(15.48)

−1

To determine α and β, we substitute (15.47) into (15.48) and integrate the right side to ﬁnd 2 2 1 2 11 2 1 4 11 +α + +β , β = +α + β . (15.49) α= 5 35 7 5 5 35 5 Solving (15.49) for α and β we obtain √ √ 24 4 35 1 35 83 6 131 (α, β) = , , ,− , ∓ ,− ± . (15.50) 35 5 70 5 140 7 5 20

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

Substituting (15.50) into (15.47) leads to the four exact solutions √ √ 5 35 35 2 2 3 2 u(x) = x + x , x − x , x± ∓ x . 2 4 7 20

475

(15.51)

Example 15.7 Use the direct computation method to solve the nonlinear Fredholm integral equation 1 2 83 50 2 u(x) = − − x + x + (1 + xt + x2 t2 )u2 (t) dt. (15.52) 15 30 21 0 We rewrite (15.52) as 83 50 2 −β x+ + γ x2 , (15.53) u(x) = − + α − 15 30 21 where 1 1 1 α= u2 (t)dt, β = tu2 (t)dt, γ = t2 u2 (t)dt. (15.54) 0

0

0

To determine α,β and γ, we substitute (15.53) into (15.54), integrate the right side of the resulting equations, and by solving the resulting system we obtain 17 23 13 , , . (15.55) (α, β, γ) = 15 30 21 Substituting (15.55) into (15.53) leads to the exact solutions u(x) = 1 − 2x + 3x2 .

(15.56)

Example 15.8 Use the direct computation method to solve the nonlinear Fredholm integral equation 1 17 2 131 691 3 − x− x −x + u(x) = (1 + x2 t + xt2 )u2 (t) dt. (15.57) 210 630 120 0 We rewrite (15.57) as 691 17 131 +α − −γ x− − β x2 − x3 , (15.58) u(x) = 210 630 120 where 1 1 1 u2 (t)dt, β = tu2 (t)dt, γ = t2 u2 (t)dt. (15.59) α= 0

0

0

To determine α,β and γ, we substitute (15.58) into (15.59), integrate the right side of the resulting equations, and by solving the resulting system we obtain 79 17 61 (α, β, γ) = , , . (15.60) 210 120 630 Substituting (15.60) into (15.58) leads to the exact solutions

476

15 Nonlinear Fredholm Integral Equations

u(x) = 1 − x − x3 .

(15.61)

Exercises 15.3.1 In Exercises 1–6, use the direct computation method to solve the following nonlinear Fredholm integral equations. Find the singular point and the bifurcation point for each equation: 1 1 tu2 (t)dt 2. u(x) = 2 + λ t2 u2 (t)dt 1. u(x) = 4 + λ 3. u(x) = 1 + λ

0

−1

5. u(x) = 3 − λ

0

1

1

0

t2 u2 (t)dt

tu2 (t)dt

√ 1 3 +λ 4. u(x) = xu2 (t)dt 4 −1 1 5 6. u(x) = x + λ t2 u2 (t)dt 4 0

In Exercises 7–12, use the direct computation method so solve the following nonlinear Fredholm integral equations: 1 1 51 2 17 x + x+ 7. u(x) = 1 − x2 u3 (t)dt 8. u(x) = 1 − x(u2 (t) + u3 (t))dt 35 3 0 −1 1 437 2 14 (xt + x2 t2 )(u(t) − u2 (t))dt x− x + 9. u(x) = 15 420 0 π 7 10. u(x) = sin x + cos(x + t)u2 (t)dt 3 0 1 11 7 163 2 11. u(x) = − + x− x + (1 + xt + x2 t2 )u2 (t)dt 15 15 105 −1 π 1 π (1 + sin(x + t))u2 (t)dt 12. u(x) = cos x − + 3 2 0

15.3.2 The Series Solution Method In this section, the series solution method will be applied to handle nonlinear Fredholm integral equations. Recall that the generic form of Taylor series at x = 0 can be written as ∞ u(x) = a n xn . (15.62) n=0

We will assume that the solution u(x) of the nonlinear Fredholm integral equation 1 u(x) = f (x) + λ K(x, t)F (u(t))dt, (15.63) 0

exists and is analytic, and therefore possesses a Taylor series of the form given in (15.62), where the coeﬃcients an will be determined recurrently. Substituting (15.62) into both sides of (15.63) gives

15.3 Nonlinear Fredholm Integral Equations of the Second Kind ∞

1

n

an x = T (f (x)) +

n=0

or for simplicity we use a0 +a1 x+a2 x2 +· · · = T (f (x))+

K(x, t)F

0

∞

477

a n tn

dt,

(15.64)

n=0

0

1

K(x, t)F (a0 +a1 t+a2 t2 +· · · )dt, (15.65)

where T (f (x)) is the Taylor series for f (x). The integral equation (15.63) will be converted to a traditional integral in (15.64) or (15.65), where instead of integrating the nonlinear term F (u(x)), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. Proceeding as in previous chapters, we ﬁrst integrate the right side of the integral in (15.64) or (15.65), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x in both sides of the resulting equation to obtain a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (15.62). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we determine, the higher accuracy level we achieve. Recall that the series solution method works eﬀectively if the solution u(x) is a polynomial. However, if u(x) is not a polynomial, then approximations to the coeﬃcients aj , j 0 will be used. Example 15.9 Solve the nonlinear Fredholm integral equation by using the series solution method 1 2309 2 1 7159 x+ x + (xt2 − x2 t)u2 (t)dt. (15.66) u(x) = 1 + 7560 2160 36 0 Using the series form (15.62) into both sides of (15.66) gives 2309 2 7159 x+ x a0 + a1 x + a2 x2 + · · · = 1 + 7560 2160 1

2 1 + (xt2 − x2 t) a0 + a1 t + a2 t2 + · · · dt, 36 0 (15.67) where by integrating the right side, collecting the like powers of x, and equating the coeﬃcients of like powers of x in both sides yields a0 = 1, 1, a1 = 1, 3611.190273, (15.68) a2 = 1, −4848.332424, an = 0, for n 3. The exact solutions are given by

478

15 Nonlinear Fredholm Integral Equations

u(x) = 1 + x + x2 , 1 + 3611.190273x − 4848.332424x2.

(15.69)

Example 15.10 Solve the nonlinear Fredholm integral equation by using the series solution method 1 41 76 2 1 u(x) = 2 − x + x − x3 + (xt − x2 t2 )u2 (t)dt. (15.70) 45 945 48 −1 Using the series form (15.62) into both sides of (15.70) gives 76 2 41 x − x3 a0 + a1 x + a2 x2 + · · · = 2 − x + 45 945 1

2 1 + (xt − x2 t2 ) a0 + a1 t + a2 t2 + · · · dt, 48 −1 (15.71) where by integrating the right side, collecting the like powers of x, and equating the coeﬃcients of like powers of x in both sides yields a0 = 2, 2, a1 = −1, 0.2924553739, a2 = 0, −173.6222619, a3 = −1, −1, (15.72) an = 0, for n 4. The exact solutions are given by (15.73) u(x) = 2 − x − x2 , 2 + 0.2924553739x − 173.6222619x2 − x3 . Example 15.11 Solve the nonlinear Fredholm integral equation by using the series solution method 1 1 1 x 2 u(x) = e + (3 − e ) + (x − t)u2 (t)dt. (15.74) 16 4 0 Substituting the series (15.62) into both sides of (15.74) gives a0 + a1 x + a2 x2 + a3 x3 + · · ·

2 1 1 1 (x − t) a0 + a1 t + a2 t2 + · · · dt. = ex + (3 − e2 ) + 16 4 0 Proceeding as before we ﬁnd 1 a0 = 1, a1 = 1, a2 = , 2! 1 1 1 a3 = , a4 = , · · · , an = . 3! 4! n! This gives the solution in a series form 1 1 1 u(x) = 1 + x + x2 + x3 + x4 + · · · . 2! 3! 4! Consequently, the exact solution is given by u(x) = ex .

(15.75)

(15.76)

(15.77) (15.78)

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

479

Example 15.12 We next consider the nonlinear Fredholm integral equation π π π 1 u(x) = cos x − sin x − (x − t)u2 (t)dt. + (1 + π) + 60 120 60 0 Substituting the series (15.62) into both sides of (15.79) gives a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · π π + (1 + π) = cos x − sin x − 60 120 π

2 1 + (x − t) a0 + a1 t + a2 t2 + · · · dt. 60 0 Proceeding as before we ﬁnd 1 a0 = 1, a1 = −1, a2 = − , 2! 1 1 1 a 3 = , a 4 = , a5 = − , · · · , 3! 4! 5! and so on. The solution in a series form is given by 1 2 1 4 1 3 1 5 u(x) = 1 − x + x + · · · − x − x + x + · · · , 2! 4! 3! 5! that converges to the exact solution u(x) = cos x − sin x.

(15.79)

(15.80)

(15.81)

(15.82)

(15.83)

Exercises 15.3.2 Solve the following nonlinear Fredholm integral equations by using the series solution method 1 1 241 1091 (x − t)u2 (t)dt 1. u(x) = − x − x2 + 240 1080 36 0 1 7537 1 119 1091 2 2. u(x) = (x − t)2 u2 (t)dt − x− x + 7560 120 1080 36 0 1 23 1 223 2 3. u(x) = 1 − (xt2 − x2 t)u2 (t)dt x+ x + 945 216 36 0 1 23 1 4. u(x) = 1 − (xt2 − x2 t)u2 (t)dt x + x2 + 630 48 −1 1 892 2357 1 + x − x2 − x3 + 5. u(x) = (x − t)u3 (t)dt 2310 945 18 0 1 1 1 1 (1 − e2 )x + (1 + e2 ) + 6. u(x) = ex + (x − t)u2 (t)dt 16 32 8 0 1 1 1 xtu2 (t)dt (1 + e2 )x + 7. u(x) = ex − 32 8 0

480

15 Nonlinear Fredholm Integral Equations

8. u(x) = cos x −

1 π + 112 56

1 0

u2 (t)dt

15.3.3 The Adomian Decomposition Method The Adomian decomposition method has been outlined before in previous chapters and has been applied to a wide class of linear Fredholm integral equations and linear Fredholm integro-diﬀerential equations. The method usually decomposes the unknown function u(x) into an inﬁnite sum of components that will be determined recursively through iterations as discussed before. The Adomian decomposition method will be applied in this chapter to handle nonlinear Fredholm integral equations. Although the linear term u(x) is represented by an inﬁnite sum of components, the nonlinear terms such as u2 , u3 , u4 , sin u, eu , etc. that appear in the equation, should be expressed by a special representation, called the Adomian polynomials An , n 0. Adomian introduced a formal algorithm to establish a reliable representation for all forms of nonlinear terms. The Adomian polynomials were introduced in Chapter 13. In what follows we present a brief outline for using the Adomian decomposition method for solving the nonlinear Fredholm integral equation b u(x) = f (x) + λ K(x, t)F (u(t))dt, (15.84) a

where F (u(t)) is a nonlinear function of u(x). The nonlinear Fredholm integral equation (15.84) contains the linear term u(x) and the nonlinear function F (u(x)). The linear term u(x) of (15.84) can be represented normally by the decomposition series ∞ un (x), (15.85) u(x) = n=0

where the components un (x), n 0 can be easily computed in a recursive manner as discussed before. However, the nonlinear term F (u(x)) of (15.84) should be represented by the so-called Adomian polynomials An by using the algorithm n 1 dn i An = λ ui , n = 0, 1, 2, · · · . (15.86) F n! dλn i=0

λ=0

Substituting (15.85) and (15.86) into (15.84) gives ∞ b ∞ un (x) = f (x) + λ K(x, t) An (t) dt. n=0

a

(15.87)

n=0

To determine the components u0 (x), u1 (x), . . ., we use the following recurrence relation

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

481

u0 (x) = f (x), b K(x, t)A0 (t)dt, u1 (x) = λ a

b

u2 (x) = λ

K(x, t)A1 (t)dt,

(15.88)

a

.. .

b

K(x, t)An (t)dt, n 0.

un+1 (x) = λ a

Recall that in the modiﬁed decomposition method, a modiﬁed recurrence relation is usually used, where f (x) is decomposed into two components f1 (x) and f2 (x), such that f (x) = f1 (x)+f2 (x). In this case the modiﬁed recurrence relation becomes in the form u0 (x) = f1 (x), b K(x, t)A0 (t)dt, u1 (x) = f2 (x) + λ

a b

u2 (x) = λ

K(x, t)A1 (t)dt,

(15.89)

a

.. .

b

K(x, t)An (t)dt, n 0.

un+1 (x) = λ a

Having determined the components, the solution in a series form is readily obtained. The obtained series solution may converge to the exact solution if such a solution exists, otherwise the series can be used for numerical purposes. The following remarks can be observed: (i) The convergence of the decomposition method has been examined in the literature by many authors. (ii) The decomposition method always gives one solution, although the solution of the nonlinear Fredholm equation is not unique. The decomposition method does not address the existence and uniqueness concepts. (iii) The modiﬁed decomposition method and the noise terms phenomenon can be used to accelerate the convergence of the solution. Generally speaking, the Adomian decomposition method is reliable and eﬀective to handle diﬀerential and integral equations. This will be illustrated by using the following examples. Example 15.13 Use the Adomian decomposition method to solve the nonlinear Fredholm integral equation

482

15 Nonlinear Fredholm Integral Equations

1

u(x) = a + λ

u2 (t)dt, a > 0.

(15.90)

0

The Adomian polynomials for the nonlinear term u2 (x) are given by A0 (x) = u20 (x), A1 (x) = 2u0 (x)u1 (x), (15.91) A2 (x) = 2u0 (x)u2 (x) + u21 (x), and so on. Substituting the series (15.85) and the Adomian polynomials (15.91) into the left side and the right side of (15.90) respectively we ﬁnd 1 ∞ ∞ un (x) = a + λ An (t)dt. (15.92) 0 n=0

n=0

Using the Adomian decomposition method we set 1 Ak (t)dt, k 0. u0 (x) = a, uk+1 (x) = λ

(15.93)

0

This in turn gives u0 (x) = a, u1 (x) = λ u2 (x) = λ u3 (x) = λ u4 (x) = λ

1 0 1 0

u20 (t)dt = λa2 , (2u0 (t)u1 (t))dt = 2λ2 a3 , (15.94)

1

(2u0 (t)u2 (t) +

0 1 0

u21 (t))dt

3 4

= 5λ a ,

(2u0 (t)u3 (t) + 2u1 (t)u2 (t))dt = 14λ4 a5 ,

.. . The solution in a series form is given by u(x) = a + λa2 + 2λ2 a3 + 5λ3 a4 + 14λ4 a5 + · · · ,

(15.95)

that converges to the exact solution √ 1 1 − 1 − 4aλ ,0 < λ . (15.96) u(x) = 2λ 4a It is clear that only one solution was obtained by using the Adomian decomposition method. However, by using the direct computation method we obtained two solutions for this nonlinear problem as shown in Example 1 that was presented before. Example 15.14 Use the Adomian decomposition method to solve the nonlinear Fredholm integral equation 1 u(x) = 1 + λ (1 − u(t) + u2 (t))dt. (15.97) 0

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

This equation can be rewritten as

u(x) = 1 + λ + λ

1

(−u(t) + u2 (t))dt.

483

(15.98)

0

Substituting the series assumption for u(x) and the Adomian polynomials for u2 (x) into the left side and the right side of (15.98) we ﬁnd 1 ∞ ∞ ∞ un (x) = 1 + λ + λ un (t) + An (t) dt. (15.99) − 0

n=0

n=0

n=0

Using the Adomian decomposition method we set 1 (−uk (t) + Ak (t))dt, k 0. u0 (x) = 1 + λ, uk+1 (x) = λ

(15.100)

0

This in turn gives u0 (x) = 1 + λ, 1 (−u0 (t) + u20 (t))dt = λ2 + λ3 , u1 (x) = λ 0 1 u2 (x) = λ (−u1 (t) + 2u0 (t)u1 (t))dt = λ3 + 3λ4 + 2λ5 , 0 1 (−u2 (t) + 2u0 (t)u2 (t) + u21 (t))dt = λ4 + 6λ5 + 10λ6 + 5λ7 , u3 (x) = λ 0 1 (−u3 (t) + 2u0 (t)u3 (t) + 2u1 (t)u2 (t))dt = λ5 + 10λ6 + · · · , u4 (x) = λ .. .

0

(15.101) The solution in a series form is given by (15.102) u(x) = 1 + λ + λ2 + 2λ3 + 4λ4 + 9λ5 + · · · , that converges to the exact solution 1 + λ − (1 − 3λ)(1 + λ) u(x) = . (15.103) 2λ It is clear again that only one solution was obtained by using the Adomian decomposition method. However, by using the direct computation method we can easily derive two solutions given by 1 + λ ± (1 − 3λ)(1 + λ) . (15.104) u(x) = 2λ It is worth noting that two bifurcation points appear at λ = −1 and λ = 13 . Example 15.15 Use the modiﬁed Adomian decomposition method to solve the nonlinear Fredholm integral equation 1 1 − ex+3 + ext u3 (t)dt. (15.105) u(x) = ex + x+3 0

484

15 Nonlinear Fredholm Integral Equations

Substituting the series (15.85) and the Adomian polynomials (15.86) into the left side and the right side of (15.105) respectively we ﬁnd 1 ∞ ∞ 1 − ex+3 x xt + un (x) = e + e An (t)dt, (15.106) x+3 0 n=0 n=0 where An are the Adomian polynomials for u2 (x) as shown above. Using the modiﬁed decomposition method we set u0 (x) = ex , 1 1 − ex+3 + ext A0 (t)dt, u1 (x) = x+3 0 1 (15.107) 1 − ex+3 + ext u30 (t)dt = 0, = x+3 0 1 ext Ak (t)dt = 0, k 1. uk+1 (x) = 0

This in turn gives the exact solution u(x) = ex ,

(15.108)

that satisﬁes the integral equation. It is worth noting that we did not use the Adomian polynomials. This is due to the fact that the modiﬁed decomposition method accelerates the convergence of the solution. Example 15.16 Use the modiﬁed Adomian decomposition method to solve the nonlinear Fredholm integral equation 1 2x(u2 (t) − tan2 (t))dt. (15.109) u(x) = sec x − 2x + 0

Substituting the series (15.85) and the Adomian polynomials (15.86) into the left side and the right side of (15.109) respectively we ﬁnd 1 ∞ ∞ 2 un (x) = sec x − 2x + xt An (t) − tan (t) dt. (15.110) 0

n=0

n=0

Using the modiﬁed decomposition method we set u0 (x) = sec x, 1 xt(A0 (t) − tan2 (t))dt = 0. u1 (x) = −2x +

(15.111)

0

Consequently, the exact solution is given by u(x) = sec x.

(15.112)

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

485

Exercises 15.3.3 In Exercises 1–6, solve the following nonlinear Fredholm integral equations by using the Adomian decomposition method 1 1 1. u(x) = 4 + λ tu2 (t)dt 2. u(x) = 2 + λ t2 u2 (t)dt 0

3. u(x) = 1 + λ

−1

5. u(x) = 3 − λ

0

1

1

0

√ 1 3 4. u(x) = xu2 (t)dt +λ 4 −1 1 5 6. u(x) = x + λ t2 u2 (t)dt 4 0

t2 u2 (t)dt

tu2 (t)dt

In Exercises 7–12, solve the nonlinear Fredholm integral equations by using the modiﬁed Adomian decomposition method π √e−1 1 2t π 7. u(x) = tan x − + dt 8. u(x) = x − 1 + dt 2 2 1 + u2 (t) 0 1 + u (t) 0 π π 4 x(u2 (t) − 1)dt 9. u(x) = sec x + −1 x+ 4 0 1 1 10. u(x) = cosh x + − x + (x − t)(u2 − sinh2 t)dt 2 0 1 1 (x − t)u2 (t)dt 11. u(x) = ln x − 2x + + 4 0 1 (x − t)(1 − u(t) + u2 (t))dt 12. u(x) = ln x − 4x + 1 + 0

15.3.4 The Successive Approximations Method The successive approximations method or the Picard iteration method was introduced before in Chapter 3. The method provides a scheme that can be used for solving initial value problems or integral equations. This method solves any problem by ﬁnding successive approximations to the solution by starting with an initial guess as u0 (x), called the zeroth approximation. As will be seen, the zeroth approximation is any selective real-valued function that will be used in a recurrence relation to determine the other approximations. The most commonly used values for the zeroth approximations are 0, 1, or x. Of course, other real values can be selected as well. Given Fredholm integral equation of the second kind b u(x) = f (x) + λ K(x, t)F (u(t))dt, (15.113) a

where u(x) is the unknown function to be determined, K(x, t) is the kernel, F (u(t)) is a nonlinear function of u(t), and λ is a parameter. The successive approximations method introduces the recurrence relation

486

15 Nonlinear Fredholm Integral Equations

u0 (x) = any selective real valued function, b K(x, t)un (t)dt, n 0. un+1 (x) = f (x) + λ

(15.114)

a

The question of convergence of un (x) for linear equation is justiﬁed by Theorem 1 presented in Chapter 3. However, for nonlinear Fredholm integral equation, it was proved in [1] that 1 , (15.115) λ< k(b − 1) R and M as shown where k is the larger of the two numbers K 1 + |λ|K(b−a) above. At the limit, the solution is determined by using the limit u(x) = lim un+1 (x). n→∞

(15.116)

The successive approximations method, or the Picard iteration method will be illustrated by studying the following examples. Example 15.17 Use the successive approximations method to solve the nonlinear Fredholm integral equation π π2 1 u(x) = cos x − tu2 (t)dt. (15.117) + 48 12 0 For the zeroth approximation u0 (x), we can select (15.118) u0 (x) = 1. The method of successive approximations admits the use of the iteration formula π 1 π2 un+1 (x) = cos x − + tu2n(t)dt, n 0. (15.119) 48 12 0 Substituting (15.118) into (15.119) we obtain π π2 1 tu20 (t)dt = cos(x) + 0.2056167584, + u1 (x) = cos x − 48 12 0 π 1 π2 + u2 (x) = cos x − tu21 (t)dt = cos(x) − 0.05115268549, 48 12 0 π 1 π2 + u3 (x) = cos x − tu22 (t)dt = cos(x) + 0.01812692764, 48 12 0 (15.120) π 2 π 1 tu23 (t)dt = cos(x) − 0.005907183842, + u4 (x) = cos x − 48 12 0 π 1 π2 + u5 (x) = cos x − tu24 (t)dt = cos(x) + 0.001983411200, 48 12 0 .. . and so on. Consequently, the solution u(x) of (15.117)

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

u(x) = lim un+1 (x) = cos x. n→∞

487

(15.121)

Unlike the direct computation method where we may get more than a solution, the successive substitution method gives only one solution for the nonlinear Fredholm problem. However, using the direct computation method gives the two solutions 32 u(x) = cos x, cos x + 2 . (15.122) π Example 15.18 Use the successive approximations method to solve the nonlinear Fredholm integral equation 1 x 2 1 x (e + 1) + xtu2 (t)dt. (15.123) u(x) = e − 192 48 0 For the zeroth approximation u0 (x), we can select (15.124) u0 (x) = 1. The method of successive approximations admits the use of the iteration formula 1 x 2 1 un+1 (x) = ex − (e + 1) + xtu2n (t)dt, n 0. (15.125) 192 48 0 Substituting (15.124) into (15.125) we obtain 1 x 2 1 x (e + 1) + xtu20 (t)dt = ex − 0.03327633383x, u1 (x) = e − 192 48 0 1 x 2 1 xtu21 (t)dt = ex − 0.0009901404827x, (e + 1) + u2 (x) = ex − 192 48 0 1 x 2 1 xtu22 (t)dt = ex − 0.00002962822377x, u3 (x) = ex − (e + 1) + 192 48 0 1 x 2 1 x u4 (x) = e − xtu23 (t)dt = ex − 0.000000886721005x, (e + 1) + 192 48 0 .. . , (15.126) and so on. Consequently, the solution u(x) of (15.123) is given by u(x) = lim un+1 (x) = ex . n→∞

(15.127)

It is worth noting that the direct computation method gives an additional solution to this equation given by u(x) = ex + (208 − 8e)x. (15.128) Example 15.19 Use the successive approximations method to solve the nonlinear Fredholm integral equation

488

15 Nonlinear Fredholm Integral Equations

π 5π 2 1 π − + t(u(t) + u2 (t))dt. 12 144 36 0 For the zeroth approximation u0 (x), we can select u(x) = sin x + 1 −

(15.129)

(15.130) u0 (x) = 1. The method of successive approximations admits the use of the iteration formula π 5π 2 1 π − + t(un (t) + u2n (t))dt, n 0. (15.131) un+1 (x) = sin x + 1 − 12 144 36 0 Substituting (15.130) into (15.131), and proceeding as before we obtain the approximations u1 (x) = sin x + 0.6696616927, u2 (x) = sin x + 0.8214573046, u3 (x) = sin x + 0.8997853785,

(15.132)

u4 (x) = sin x + 0.9426743063, u5 (x) = sin x + 0.9668710030, and so on. Consequently, the solution u(x) of (15.129) is given by u(x) = lim un+1 (x) = 1 + sin x. n→∞

(15.133)

The direct computation method gives an additional solution to this equation given by 18 4 1− . (15.134) u(x) = sin x − 2 − π π Example 15.20 Use the successive approximations method to solve the nonlinear Fredholm integral equation 1 143 1 tu2 (t)dt. (15.135) u(x) = ln x + + 144 36 0 For the zeroth approximation u0 (x), we can select u0 (x) = 1. (15.136) The method of successive approximations admits the use of the iteration formula 1 1 143 + un+1 (x) = ln x + tu2n (t)dt, n 0. (15.137) 144 36 0 Substituting (15.136) into (15.137), and proceeding as before we obtain the approximations

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

u1 (x) = u2 (x) = u3 (x) = u4 (x) = u5 (x) = .. .

ln x + 1.006944444, ln x + 1.000097120, ln x + 1.000001349, ln x + 1.000000019, ln x + 1.000000000,

489

(15.138)

and so on. Consequently, the solution u(x) of (15.135) u(x) = lim un+1 (x) = 1 + ln x. n→∞

(15.139)

It is worth noting that the direct computation method gives an additional solution to this equation given by u(x) = 72 + ln x.

(15.140)

Exercises 15.3.4 Use the successive approximations method to solve the following nonlinear Fredholm integral equations: π 1 π2 t(1 + u2 (t))dt + 1. u(x) = sin x − 64 48 0 π 5π 1 π 1+ + 2. u(x) = sin x + 1 − t(u + u2 (t))dt 16 12 48 0 π 5π 2 1 7 + 3. u(x) = cos x + − t(u + u2 (t))dt 6 144 36 0 1 1 1 t(u + u2 (t))dt (127 − e2 ) + 4. u(x) = ex + 144 36 0 1 1 1 5. u(x) = ex + t(1 + u2 (t))dt (131 − e2 ) + 144 36 0 1 1 1 6. u(x) = xex − xtu2 (t)dt (3 + e2 )x + 288 36 0 1 1 1 7. u(x) = ex + xt2 u2 (t)dt (1 − e2 )x + 384 96 0 1 1 1 8. u(x) = ex + u3 (t)dt (1 − e3 ) + 288 96 0 π 1 1 2 x+ 9. u(x) = cos x − xu3 (t)dt 144 96 0 1 1 1 1 x+ + 10. u(x) = ln x − (x − t)u2 (t)dt 18 144 36 0 1 1 10279 tu2 (t)dt x+ 11. u(x) = x ln x + 10368 36 0

490 12. u(x) = x + ln x −

15 Nonlinear Fredholm Integral Equations 1 5 + 648 36

1 0

tu2 (t)dt

15.4 Homogeneous Nonlinear Fredholm Integral Equations Substituting f (x) = 0 into the nonlinear Fredholm integral equation of the second kind b K(x, t)F (u(t))dt, (15.141) u(x) = f (x) + λ a

gives the homogeneous nonlinear Fredholm integral equation of the second kind given by b u(x) = λ K(x, t)F (u(t))dt. (15.142) a

In this section we will focus our study on the homogeneous nonlinear Fredholm integral equation (15.142) for the speciﬁc case where the kernel K(x, t) is separable. The main goal for studying the homogeneous nonlinear Fredholm equation is to ﬁnd nontrivial solution, because the trivial solution u(x) = 0 is a solution of this equation. Moreover, the Adomian decomposition method is not applicable here because it depends mainly on assigning a non-zero value for the zeroth component u0 (x), and f (x) = 0 in this kind of equations. The direct computation method will be appropriate to be employed here to handle this kind of equations.

15.4.1 The Direct Computation Method The direct computation method was used before in this chapter. This method replaces the homogeneous nonlinear Fredholm integral equations by a single algebraic equation or by a system of simultaneous algebraic equations depending on the number of terms of the separable kernel K(x, t). As stated before, the direct computation method handles Fredholm integral equations, homogeneous or nonhomogeneous, in a direct manner and gives the solution in an exact form and not in a series form. The direct computation method will be applied in this section for the degenerate or separable kernels of the form n gk (x)hk (t). (15.143) K(x, t) = k=1

15.4 Homogeneous Nonlinear Fredholm Integral Equations

491

The direct computation method can be applied as follows: 1. We ﬁrst substitute (15.143) into the homogeneous nonlinear Fredholm integral equation the form b u(x) = λ K(x, t) F (u(t))dt. (15.144) a

2. This substitution leads to b b u(x) = λg1 (x) h1 (t) F (u(t))dt + λg2 (x) h2 (t) F (u(t))dt + · · · a a b hn (t) F (u(t))dt. +λgn (x) a

(15.145) 3. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Consequently, Equation (15.145) becomes u(x) = λα1 g1 (x) + λα2 g2 (x) + · · · + λαn gn (x), where

(15.146)

b

hi (t) F (u(t))dt, 1 i n.

αi =

(15.147)

a

4. Substituting (15.146) into (15.147) gives a system of n simultaneous algebraic equations that can be solved to determine the constants αi , 1 i n. Using the obtained numerical values of αi into (15.146), the solution u(x) of the homogeneous nonlinear Fredholm integral equation (15.142) follows immediately. Example 15.21 Solve the homogeneous nonlinear Fredholm integral equation by using the direct computation method π2 cos x sin tu2 (t)dt. (15.148) u(x) = λ 0

This equation can be rewritten as u(x) = αλ cos x, where

π 2

α=

sin tu2 (t)dt.

(15.149) (15.150)

0

Substituting (15.149) into (15.150) gives π2 sin t cos2 tdt, α = α2 λ2

(15.151)

0

that gives 1 2 2 α λ . (15.152) 3 Recall that α = 0 gives the trivial solution. For α = 0, we ﬁnd that α=

492

15 Nonlinear Fredholm Integral Equations

3 . λ2 This in turn gives the eigenfunction u(x) by 3 u(x) = cos x. λ Notice that λ = 0 is a singular point. α=

(15.153)

(15.154)

Example 15.22 Solve the homogeneous nonlinear Fredholm integral equation by using the direct computation method 1 u(x) = λ ex−tu2 (t)dt. (15.155) 0

This equation can be rewritten as u(x) = αλex , where

1

(15.156)

e−t u2 (t)dt.

(15.157)

Substituting (15.156) into (15.157) gives 1 2 2 et dt, α=α λ

(15.158)

α= 0

0

that gives

α = α2 λ2 (e − 1).

(15.159)

Recall that α = 0 gives the trivial solution. For α = 0, we ﬁnd that 1 . (15.160) α= 2 λ (e − 1) This in turn gives the eigenfunction u(x) by 1 u(x) = (15.161) ex . λ(e − 1) Example 15.23 Solve the homogeneous nonlinear Fredholm integral equation by using the direct computation method π2 u(x) = λ sin(x − 2t)u2 (t)dt. (15.162) 0

Notice that the kernel sin(x − 2t) = sin x cos 2t − cos x sin 2t is separable. Equation (15.162) can be rewritten as u(x) = αλ sin x − βλ cos x, (15.163) where

α= 0

π 2

cos 2tu2 (t)dt,

π 2

β=

sin 2tu2 (t)dt.

(15.164)

0

Substituting (15.163) into (15.164) and proceeding as before we obtain

15.4 Homogeneous Nonlinear Fredholm Integral Equations

493

λ2 π 2 λ2 (α − β 2 ), β = − (αβπ − 2α2 − 2β 2 ). 8 4 Solving this system for α and β gives 32 8π , β=− 2 2 . α=− 2 2 λ (π − 16) λ (π − 16) This in turn gives the eigenfunction u(x) by 8 u(x) = − (π sin x − 4 cos x). λ(π 2 − 16) α=−

(15.165)

(15.166)

(15.167)

Example 15.24 Solve the homogeneous nonlinear Fredholm integral equation by using the direct computation method 1 u(x) = λ (xt + t2 )(u(t) + u2 (t))dt. (15.168) −1

Equation (15.168) can be rewritten as u(x) = αλx + βλ, where

1

t(u(t) + u2 (t))dt,

α=

β=

−1

1

(15.169)

t2 (u(t) + u2 (t))dt.

(15.170)

0

Proceeding as before we ﬁnd * 5 2λ − 3 2λ − 3 2λ − 3 , β=− α = 0, ,− . 2 3 4λ 2λ2 4λ2 This in turn gives the exact solutions √ 2λ − 3 (2λ − 3)( 15x − 3) , . u(x) = − 2λ 12λ

(15.171)

(15.172)

Exercises 15.4.1 Use the direct computation method to solve the homogeneous nonlinear Fredholm integral equations π π 2 2 sin x cos tu2 (t)dt 2. u(x) = λ sin x cos t(u(t) + u2 (t))dt 1. u(x) = λ 3. u(x) = λ 5. u(x) = λ 7. u(x) = λ 9. u(x) = λ

0

π 0 1 0 π 2

0 π 0

cos x cos t(t + u2 (t))dt

4. u(x) = λ

ex−2t u3 (t)dt

6. u(x) = λ

cos(x − 2t)u2 (t)dt

8. u(x) = λ

cos(x + 2t)(1 + u2 (t))dt

10. u(x) = λ

0

1

ex−t (u(t) + u2 (t))dt

0 1

ex−2t (3t2 + e−2t u2 (t))dt

0 π 2

0

sin(x + t)u2 (t)dt

1

−1

(xt + t2 )u2 (t)dt

494

15 Nonlinear Fredholm Integral Equations

11. u(x) = λ

1 −1

(xt + x2 t2 )u2 (t)dt

12. u(x) = λ

1

0

(xt − xt2 )u2 (t)dt

15.5 Nonlinear Fredholm Integral Equations of the First Kind The standard form of the nonlinear Fredholm integral equations of the ﬁrst kind is given by b f (x) = K(x, t)F (u(t))dt, (15.173) a

where the kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x). The linear Fredholm integral equation of the ﬁrst kind is presented in Chapter 4 where the homotopy perturbation method was used for handling this type of equations. To determine a solution for the nonlinear Fredholm integral equation of the ﬁrst kind (15.173), we ﬁrst convert it to a linear Fredholm integral equation of the ﬁrst kind of the form b f (x) = K(x, t)v(t)dt, x ∈ D (15.174) a

by using the transformation v(x) = F (u(x)). We assume that F (u(x)) is invertible, then we can set

(15.175)

(15.176) u(x) = F −1 (v(x)). The linear Fredholm integral equation of the ﬁrst kind has been investigated in Chapter 4. An important remark has been reported in [3] and other references concerning the data function f (x). The function f (x) must lie in the range of the kernel K(x, t) [3]. For example, if we set the kernel by (15.177) K(x, t) = ex sin t. Then if we substitute any integrable function F (u(x)) in (15.173), and we evaluate the integral, the resulting f (x) must clearly be a multiple of ex [3]. This means that if f (x) is not a multiple of the x component of the kernel, then a solution for (15.173) does not exist. This necessary condition on f (x) can be generalized. In other words, the data function f (x) must contain components which are matched by the corresponding x components of the kernel K(x, t) Nonlinear Fredholm integral equation of the ﬁrst kind is considered illposed problem because it does not satisfy the following three properties: 1. Existence of a solution. 2. Uniqueness of a solution.

15.5 Nonlinear Fredholm Integral Equations of the First Kind

495

3. Continuous dependence of the solution u(x) on the data f (x). This property means that small errors in the data f (x) should cause small errors [4] in the solution u(x). The three properties were postulated by Hadamard [5]. Any problem that satisﬁes the three aforementioned properties is called well-posed problem. For any ill-posed problem, a very small change on the data f (x) can give a large change in the solution u(x). This means that nonlinear Fredholm integral equation of the ﬁrst kind may lead to a lot of diﬃculties. Several methods have been used to handle the linear and the nonlinear Fredholm integral equations of the ﬁrst kind. The Legendre wavelets, the augmented Galerkin method, and the collocation method are examples of the methods used to handle this equation. The methods that we used so far in this text cannot handle this kind of equations independently if it is expressed in its standard form (15.174). However, in this text, we will ﬁrst apply the method of regularization that received a considerable amount of interest, especially in solving ﬁrst order integral equations. We will second apply the homotopy perturbation method [6] to handle speciﬁc cases of the Fredholm integral equations where the kernel K(x, t) is separable. In what follows we will present a brief summary of the method of regularization and the homotopy perturbation method that will be used to handle the Fredholm integral equations of the ﬁrst kind.

15.5.1 The Method of Regularization The method of regularization was established independently by Phillips [7] and Tikhonov [8]. The method of regularization consists of replacing ill-posed problem by well-posed problem. The method of regularization transforms the linear Fredholm integral equation of the ﬁrst kind b K(x, t)v(t)dt, x ∈ D, (15.178) f (x) = a

to the approximation Fredholm integral equation b μvμ (x) = f (x) − K(x, t)vμ (t)dt, x ∈ D,

(15.179)

a

where μ is a small positive parameter. It is clear that (15.179) is a Fredholm integral equation of the second kind that can be rewritten 1 1 b K(x, t)vμ (t) dt, x ∈ D. (15.180) vμ (x) = f (x) − μ μ a

496

15 Nonlinear Fredholm Integral Equations

Moreover, it was proved in [1,3,9] that the solution vμ of Equation (15.180) converges to the solution v(x) of (15.178) as μ → 0 according to the following Lemma [10]: Lemma 15.1 Suppose that the integral operator of (15.178) is continuous and coercive in the Hilbert space where f (x), u(x), and vμ (x) are deﬁned, then: 1. |vμ | is bounded independently of μ, and 2. |vμ (x) − v(x)| → 0 when μ → 0. The proof of this lemma can be found in [3, 9–10]. In summary, by combining the method of regulariztion with any of the methods used before for solving Fredholm integral equation of the second kind, we can solve Fredholm integral equation of the ﬁrst kind (15.178). The method of regulariztion transforms the ﬁrst kind equation to a second kind equation. The resulting integral equation (15.180) can be solved by any method that was presented before in this chapter. The exact solution v(x) of (15.178) can thus be obtained by v(x) = lim vμ (x). (15.181) μ→0

In what follows we will present four illustrative examples where we will use the method of regulariztion to transform the ﬁrst kind integral equation to a second kind integral equation. The resulting equation will be solved by any appropriate method that we used before. Example 15.25 Combine the method of regulariztion and the direct computation method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 12 2ex−4t u2 (t)dt. (15.182) ex = 0

We ﬁrst set

v(x) = u2 (x), u(x) = ±

to carry out (15.182) into

x

e =

1 2

v(x),

2ex−4t v(t)dt.

(15.183)

(15.184)

0

Using the method of regularization, Equation (15.184) can be transformed to 1 1 x 1 2 x−4t vμ (x) = e − 2e vμ (t) dt. (15.185) μ μ 0 To use the direct computation method, Equation (15.185) can be written as α x 1 − e , (15.186) vμ (x) = μ μ where

15.5 Nonlinear Fredholm Integral Equations of the First Kind

1 2

α= 0

2e−4t vμ (t) dt.

497

(15.187)

To determine α, we substitute (15.186) into (15.187) to ﬁnd 1 2 1 α 2e−3t dt. α= − μ μ 0 Integrating the right side and solve to ﬁnd that

(15.188)

1

α= This in turn gives 1 vμ (x) = μ

2(1 − e 2 ) 1

2 − (2 + μ)e 2

.

(15.189)

1

1−

2(1 − e 2 ) 1

2 − (2 + μ)e 2

ex .

(15.190)

The exact solution v(x) of (15.185) can be obtained by 1

v(x) = lim vμ (x) =

ex+ 2

. 2(e − 1) Using (15.183) gives the exact solution of (15.182) by , 1 ex+ 2 . u(x) = ± 1 2(e 2 − 1) Two more solutions to Equation (15.182) are given by μ→0

u(x) = ±e2x .

1 2

(15.191)

(15.192)

(15.193)

Example 15.26 Combine the method of regulariztion and the direct computation method to solve the nonlinear Fredholm integral equation of the ﬁrst kind π π sin(x − t)u2 (t)dt. (15.194) sin x = 2 0 We ﬁrst set v(x) = u2 (x), u(x) = ± v(x), (15.195) to carry out (15.194) into π π sin x = sin(x − t)v(t)dt. (15.196) 2 0 Using the method of regularization, Equation (15.196) can be transformed to π 1 π sin(x − t)vμ (t) dt. (15.197) sin x − vμ (x) = 2μ μ 0 The resulting Fredholm integral equation of the second kind will be solved by the direct computation method. Equation (15.197) can be written as α β π vμ (x) = − sin x + cos x, (15.198) 2μ μ μ where

498

15 Nonlinear Fredholm Integral Equations

π

α= 0

cos t vμ (t) dt,

β= 0

1 2

sin t vμ (t) dt.

(15.199)

To determine α and β, we substitute (15.198) into (15.199), integrate the resulting integral and solve to ﬁnd that π3 π2 μ α= , β = . (15.200) 2(π 2 + 4μ2 ) π 2 + 4μ2 Substituting this result into (15.198) gives the approximate solution 2πμ π2 sin x + 2 cos x. (15.201) vμ (x) = 2 2 π + 4μ π + 4μ2 The exact solution v(x) of (15.197) can be obtained by v(x) = lim vμ (x) = cos x. μ→0

Using (15.195) gives the exact solution of (15.194) by √ u(x) = ± cos x.

(15.202)

(15.203)

Example 15.27 Combine the method of regulariztion and the Adomian decomposition method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 1 64 xtu4 (t)dt. (15.204) x= 15 −1 We ﬁrst set v(x) = u4 (x), u(x) = ± 4 v(x), (15.205) to carry out (15.204) into

1 64 x= xtv(t)dt. (15.206) 15 −1 Using the method of regularization, Equation (15.206) can be transformed to 1 1 64 x− xtvμ (t) dt. (15.207) vμ (x) = 15μ μ −1 The resulting Fredholm integral equation of the second kind will be solved by the Adomian decomposition method, where we ﬁrst set ∞ vμ (x) = vμn (x), (15.208) n=0

and the recurrence relation 64 1 1 x, vμk+1 (x) = − vμ0 (x) = xtvμk (t) dt, k 0. 15μ μ −1 This in turn gives the components 64 128 x, x, vμ1 (x) = − vμ0 (x) = 15μ 45μ2 256 512 x, vμ3 (x) = − x, vμ2 (x) = 135μ3 405μ4

(15.209)

(15.210)

15.5 Nonlinear Fredholm Integral Equations of the First Kind

499

and so on. Substituting this result into (15.208) gives the approximate solution 64 vμ (x) = x. (15.211) 5(2 + 3μ) The exact solution v(x) of (15.207) can be obtained by 32 v(x) = lim vμ (x) = x. (15.212) μ→0 5 Using (15.205) gives the exact solution * 4 32 x. (15.213) u(x) = ± 5 It is interesting to point out that there are two more solutions to this equation given by u(x) = ±(1 + x). (15.214) Example 15.28 Combine the method of regulariztion and the successive approximations method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 1 1 xtu3 (t)dt. (15.215) x= 5 0 We ﬁrst set v(x) = u3 (x), u(x) = ± 3 v(x), (15.216) to carry out (15.215) into

1 1 xtv(t)dt. (15.217) x= 5 0 Using the method of regularization, Equation (15.215) can be transformed to 1 1 1 xt vμ (t) dt. (15.218) vμ (x) = x− 5μ μ 0 To use the successive approximations method, we ﬁrst select uμ0 (x) = 0. Consequently, we obtain the following approximations 1 x, vμ0 (x) = 0, vμ1 (x) = 5μ 1 1 x, x− vμ2 (x) = 5μ 15μ2 (15.219) 1 1 1 x + x, vμ3 (x) = x− 5μ 15μ2 45μ3 1 1 1 1 vμ4 (x) = x− x+ x− x, 5μ 15μ2 45μ3 135μ4 and so on. Based on this we obtain the approximate solution 3 x. (15.220) vμ (x) = 5(1 + 3μ) The exact solution v(x) of (15.218) can be obtained by

500

15 Nonlinear Fredholm Integral Equations

v(x) = lim uμ (x) = μ→0

3 x. 5

Using (15.216) the exact solution is given by * 3 3 u(x) = x. 5 Another solution to this equation is given by u(x) = x.

(15.221)

(15.222) (15.223)

Exercises 15.5.1 Combine the regularization method with any method to solve the nonlinear Fredholm integral equations of the ﬁrst kind 1 1 1 3 1. e2x = e2x−3t u3 (t)dt 2. e−x = et−x u3 (t)dt 3 0 0 1 1 1 10 2 x2 t2 u3 (t)dt 4. xtu2 (t)dt 3. − x = x= 27 32 0 0 1 1 32 2 29 xtu2 (t)dt 6. x2 t2 u2 (t)dt 5. x= x = 36 63 0 −1 1 1 466 2 8 7. x2 t2 u2 (t)dt 8. xtu2 (t)dt x = x= 315 35 −1 −1 π π π π 9. cos x = cos(x − t)u2 (t)dt 10. sin x = cos(x − t)u2 (t)dt 2 2 0 0 π 1 22 π 709 sin(x − t)u2 (t)dt 12. cos(x − t)u3 (t)dt 11. − cos x = x− = 2 3 720 0 0

15.5.2 The Homotopy Perturbation Method In what follows we present the homotopy perturbation method for handling the nonlinear Fredholm integral equations of the ﬁrst kind of the form b K(x, t)v(t)dt. (15.224) f (x) = a

We ﬁrst deﬁne the operator

b

L(u) = f (x) −

K(x, t)u(t)dt = 0.

(15.225)

a

We next construct a convex homotopy of the form H(u, p) = (1 − p)u(x) + pL(u)(x) = 0.

(15.226)

The embedding parameter p monotonically increases from 0 to 1. The homotopy perturbation method admits the use of the expansion

15.5 Nonlinear Fredholm Integral Equations of the First Kind

u=

∞

p n un ,

501

(15.227)

n=0

and consequently v(x) = lim

∞

p→1

pn un (x).

(15.228)

n=0

The series (15.228) converges to the exact solution if such a solution exists. Substituting (15.227) into (15.226), and proceeding as in Chapter 4 we obtain the recurrence relation u0 (x) = 0, u1 (x) = f (x), b (15.229) K(x, t)un (t)dt, n 1. un+1 (x) = un (x) − a

If the kernel is separable, i.e. K(x, t) = g(x)h(t), then the following condition b K(t, t)dt < 1, (15.230) 1 − a must be justiﬁed for convergence. We will concern ourselves only on the case where K(x, t) = g(x)h(t). The HPM will be used to solve the following nonlinear Fredholm integral equations of the ﬁrst kind. Example 15.29 Use the homotopy perturbation method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 1 ex = ex−2t u2 (t)dt. (15.231) 0

We ﬁrst set

v(x) = u2 (x), u(x) = ±

to carry out (15.231) into

ex =

1

v(x),

ex−2t v(t)dt.

(15.232)

(15.233)

0

Notice that

1 −

0

1

K(t, t)dt = 0.3678 < 1.

Using the recurrence relation (15.229) we ﬁnd v0 (x) = 0, v1 (x) = ex , 1 vn+1 (x) = vn (x) − ex−2t vn (t)dt, n 1. 0

This in turn gives

(15.234)

(15.235)

502

15 Nonlinear Fredholm Integral Equations

v1 (x) = ex , 13 v2 (x) = v1 (x) − ex−2t v1 (t)dt = ex−1 , v0 (x) = 0,

v3 (x) = v2 (x) − v4 (x) = v3 (x) − v5 (x) = v4 (x) −

0

1 3

0 1 3

0 1 3

0

ex−2t v2 (t)dt = ex−2 ,

(15.236)

ex−2t v3 (t)dt = ex−3 , ex−2t v4 (t)dt = ex−4 ,

and so on. Consequently, the approximate solution is given by v(x) = ex (1 + e−1 + e−2 + e−3 + · · · ),

(15.237)

that converges to v(x) =

ex+1 . e−1

(15.238)

Using (15.232) gives the exact solution *

ex+1 . (15.239) e−1 It is worthnoting that there are two more solutions to this equation given by u(x) = ±ex . (15.240) u(x) = ±

The reason that the solution is not unique is due to the fact that the problem is nonlinear and ill-posed as well. Example 15.30 Use the homotopy perturbation method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 12 1 −x e = et−x u2 (t)dt. (15.241) 2 0 We ﬁrst set (15.242) v(x) = u2 (x), u(x) = ± v(x), to carry out (15.241) into 1 −x e = 2 Notice that

1 2

et−x v(t)dt.

(15.243)

0

12 K(t, t)dt = 0.5 < 1. 1 − 0

Using the recurrence relation (15.229) we ﬁnd

(15.244)

15.5 Nonlinear Fredholm Integral Equations of the First Kind

1 v1 (x) = e−x , 2 1 et−x vn (t)dt, n 1. vn+1 (x) = vn (x) −

503

v0 (x) = 0,

(15.245)

0

This in turn gives v0 (x) = 0,

1 −x e , 2

v1 (x) =

v2 (x) = v1 (x) −

1 3

0

et−x v1 (t)dt =

1 3

1 −x e , 4

1 −x e , 8 0 and so on. Consequently, the approximate solution is given by 1 1 1 1 −x v(x) = e + + + + ··· , 2 4 4 16 that converges to v(x) = e−x . v3 (x) = v2 (x) −

(15.246)

ex−2t v2 (t)dt =

(15.247) (15.248)

Using (15.242) gives the exact solution 1

u(x) = ±e− 2 x .

(15.249)

Example 15.31 Use the homotopy perturbation method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 1 3 − x= xtu3 (t)dt. (15.250) 8 0 We ﬁrst set 1 (15.251) v(x) = u3 (x), u(x) = v 3 (x), to carry out (15.250) into 3 − x= 8 Notice that

1 −

1 0

1

xtv(t)dt.

2 K(t, t)dt = < 1. 3

Using the recurrence relation (15.229) we ﬁnd 3 v0 (x) = 0, v1 (x) = − x, 8 vn+1 (x) = vn (x) − This in turn gives

(15.252)

0

1

0

xtvn (t)dt, n 1.

(15.253)

(15.254)

504

15 Nonlinear Fredholm Integral Equations

3 v1 (x) = − x, 8 1 1 v2 (x) = v1 (x) − xt v1 (t)dt = − x, 4 0 (15.255) 1 1 v3 (x) = v2 (x) − xt v2 (t)dt = − x, 6 0 1 1 xt v3 (t)dt = − x, v4 (x) = v3 (x) − 9 0 and so on. Consequently, the approximate solution is given by 2 4 8 3 + ··· , (15.256) v(x) = − x 1 + + + 8 3 9 27 that converges to 9 v(x) = − x. (15.257) 8 Using (15.251) gives the exact solution * 3 9 (15.258) − x. u(x) = 8 It is interesting to point out that there is one more solution to this equation given by u(x) = ln x. (15.259) v0 (x) = 0,

The reason that the solution is not unique is due to the fact that the problem is nonlinear and ill-posed as well. Example 15.32 Use the homotopy perturbation method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 1 127 2 x = x2 t2 u2 (t)dt. (15.260) 252 0 We ﬁrst set (15.261) v(x) = u2 (x), u(x) = v(x), to carry out (15.260) into 127 2 x = 252 Notice that

1 −

0

1

1

x2 t2 v(t)dt.

(15.262)

0

2 K(t, t)dt = < 1. 3

(15.263)

Using the recurrence relation (15.229) and proceeding as before we ﬁnd

15.6 Systems of Nonlinear Fredholm Integral Equations

505

127 2 x , v1 (x) = 252 1 127 2 x , xtv1 (t)dt = v2 (x) = v1 (x) − 315 0 1 (15.264) 508 2 x , v3 (x) = v2 (x) − xtv2 (t)dt = 1575 0 1 2032 2 x , v4 (x) = v3 (x) − xtv3 (t)dt = 7875 0 and so on. Consequently, the approximate solution is given by 4 16 64 127 2 x 1+ + + + ··· , (15.265) v(x) = 252 5 25 125 that converges to 635 2 v(x) = x . (15.266) 252 Using (15.261) gives the exact solution * 635 x. (15.267) u(x) = ± 252 It is interesting to point out that there are two more solutions to this equation given by (15.268) u(x) = ±(x2 + x3 ). v0 (x) = 0,

The reason that the solution is not unique is due to the fact that the problem is nonlinear and ill-posed as well. Exercises 15.5.2 Use the homotopy perturbation method to solve the nonlinear Fredholm integral equations of the ﬁrst kind 1 1 1 3 1. e2x = e2x−3t u3 (t) dt 2. e−x = et−x u3 (t) dt 3 0 0 1 1 1 10 2 x2 t2 u3 (t) dt 4. xt u2 (t) dt 3. − x = x= 27 32 0 0 1 1 32 2 29 xt u2 (t) dt 6. x2 t2 u2 (t) dt 5. x= x = 36 63 0 −1 1 1 466 2 8 7. x2 t2 u2 (t) dt 8. xt u2 (t) dt x = x= 315 35 −1 −1

15.6 Systems of Nonlinear Fredholm Integral Equations In this section, we will study systems of nonlinear Fredholm integral equations of the second kind given by

506

15 Nonlinear Fredholm Integral Equations

b

˜ 1 (x, t)F˜1 (v(t)) dt, K1 (x, t)F1 (u(t)) + K

b

˜ 2 (x, t)F˜2 (v(t)) dt. K2 (x, t)F2 (u(t)) + K

u(x) = f1 (x) + a

v(x) = f2 (x) +

(15.269)

a

The unknown functions u(x) and v(x), that will be determined, occur inside ˜ i (x, t), and the funcand outside the integral sign. The kernels Ki (x, t) and K tion fi (x) are given real-valued functions, for i = 1, 2. The functions Fi and F˜i , for i = 1, 2 are nonlinear functions of u(x) and v(x). Systems of linear Fredholm integral equations were presented in Chapter 11. In this chapter, the system of nonlinear Fredholm integral equations can be handled by four distinct methods, namely the direct computation method, the modiﬁed Adomian method, the successive approximations method, and the series solution method. Although the aforementioned methods work effectively for handling the systems of nonlinear Fredholm integral equations, but only the ﬁrst two methods will be used in this section.

15.6.1 The Direct Computation Method The direct computation method will be applied to solve the systems of nonlinear Fredholm integral equations of the second kind. The method approaches Fredholm integral equations in a direct manner and gives the solution in an exact form and not in a series form. In what follows we summarize the necessary steps needed to apply this method. The method will be applied for the degenerate or separable kernels of the form n n ˜ k (t), ˜ 1 (x, t) = K1 (x, t) = gk (x)hk (t), K g˜k (x)h K2 (x, t) =

k=1 n

k=1

˜ 2 (x, t) = K

rk (x)sk (t),

k=1

n

(15.270) r˜k (x)˜ sk (t).

k=1

The direct computation method can be applied as follows: 1. We ﬁrst substitute (15.270) into the system (15.269) to obtain b n u(x) = f1 (x) + gk (x) hk (t)F1 (u(t)) dt +

n

k=1

k=1

˜ k (t)F˜1 (v(t)) dt, h

a

v(x) = f2 (x) + +

b

g˜k (x)

k=1

n

a

n k=1

(15.271)

b

rk (x)

sk (t)F2 (u(t)) dt a

b

s˜k (t)F˜2 (v(t)) dt.

r˜k (x) a

15.6 Systems of Nonlinear Fredholm Integral Equations

507

2. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (15.271) becomes u(x) = f1 (x) + α1 g1 (x) + · · · + αn gn(x) + β1 g˜1 (x) + · · · + βn g˜n (x), v(x) = f2 (x) + γ1 r1 (x) + · · · + γn rn (x) + δ1 r˜1 (x) + · · · + δn r˜n (x), (15.272) where b

hi (t) F1 (u(t)) dt, 1 i n,

αi = a

b

βi =

˜ i (t) F%1 (v(t))dt, 1 i n, h

a

(15.273)

b

si (t) F2 (u(t))dt, 1 i n,

γi = a

δi =

b

s˜i (t) F%2 (v(t))dt, 1 i n.

a

3. Substituting (15.272) into (15.273) gives a system of n algebraic equations that can be solved to determine the constants αi , βi , γi , and δi . To facilitate the computational work, we can use the computer symbolic systems such as Maple and Mathematica. Using the obtained numerical values of these constants into (15.272), the solutions u(x) and v(x) of the system of nonlinear Fredholm integral equations (15.269) follow immediately. The analysis presented above can be explained by studying the following examples. Example 15.33 Solve the following system of nonlinear Fredholm integral equations by using the direct computation method π u(x) = sin x + (1 − 2π) cos x + cos x(u2 (t) + v 2 (t)) dt, 0 (15.274) π (u2 (t) − v 2 (t)) dt. v(x) = sin x − cos x + 0

This system can be rewritten as u(x) = sin x + (1 − 2π + α + β) cos x, v(x) = sin x − cos x + (α − β), where π π u2 (t) dt,

α= 0

v 2 (t) dt.

β=

(15.275)

(15.276)

0

To determine α, and β, we substitute (15.275) into (15.276), and solving the resulting system, we obtain α = π, β = π. (15.277)

508

15 Nonlinear Fredholm Integral Equations

Substituting this result into (15.275) leads to the exact solutions (u(x), v(x)) = (sin x + cos x, sin x − cos x).

(15.278)

Example 15.34 Solve the following system of nonlinear Fredholm integral equations by using the direct computation method π4 π (tan x u2 (t) + sec x v 2 (t)) dt, u(x) = sec x − tan x + 4 0 (15.279) π4 π 2 2 v(x) = 1 − sec x + (tan x u (t) − sec x v (t)) dt. 4 0 Following the analysis presented above this system can be rewritten as π + β sec x + (α − 1) tan x, u(x) = 4 (15.280) π v(x) = 1 − − β sec x + α tan x, 4 where π4 π4 2 α= u (t) dt, β = v2 (t) dt. (15.281) 0

0

To determine α and β, we substitute (15.280) into (15.281) and solve the resulting system, we ﬁnd π α = 1, β = 1 − . (15.282) 4 Substituting (15.282) into (15.280) leads to the exact solutions (u(x), v(x)) = (sec x, tan x).

(15.283)

Example 15.35 Solve the following system of nonlinear Fredholm integral equations by using the direct computation method 1 x(u2 (t) + v 2 (t)) dt, u(x) = 1 − 6x + ln x + + 0 (15.284) 1 v(x) = 1 + 4x2 − ln x +

0+

x2 (u2 (t) − v 2 (t)) dt.

Proceeding as before, this system can be rewritten as u(x) = 1 + (α + β − 6)x + ln x, v(x) = 1 + (4 + α − β)x2 − ln x, where 1

α= 0+

u2 (t) dt,

1

β=

v2 (t) dt.

(15.285)

(15.286)

0+

To determine α, and β, we substitute (15.285) into (15.286), and proceeding as before to obtain α = 1, β = 5. (15.287)

15.6 Systems of Nonlinear Fredholm Integral Equations

509

This in turn gives the exact solutions (u(x), v(x)) = (1 + ln x, 1 − ln x).

(15.288)

Example 15.36 Solve the following system of nonlinear Fredholm integral equations by using the direct computation method π4 2 35(v 2 (t) − w2 (t)) dt, u(x) = 27 + sec x + 0

v(x) = 115 − sec2 x + w(x) = −3 − sec4 x +

π 4

0

0

π 4

35(w2 (t) − u2 (t)) dt, (u2 (t) − v 2 (t)) dt.

Proceeding as before, this system can be rewritten as u(x) = 27 + sec2 x + 35(β − γ), v(x) = 115 − sec2 x + 35(γ − α), w(x) = −3 − sec4 x + α − β, where π4 π4 π4 α= u2 (t) dt, β = v 2 (t) dt, γ = w2 (t)dt. 0

(15.289)

0

(15.290)

(15.291)

0

To determine α, β, and γ, we substitute (15.290) into (15.291) and by solving the resulting system we ﬁnd 10 π 2 π 8 π . (15.292) α= + , β= − , γ= + 4 3 4 3 4 105 These results lead to the exact solutions (u(x), v(x), w(x)) = (1 + sec2 x, 1 − sec2 x, 1 − sec4 x).

(15.293)

Exercises 15.6.1 Use the direct computation method to solve the following systems of nonlinear Fredholm 1 ⎧ integral equations

4 ⎪ ⎪ u(x) = x − + t u2 (t) + v 2 (t) dt ⎪ ⎨ 7 −1 1. 1 ⎪

4 ⎪ ⎪ ⎩ v(x) = x2 + x3 + + t u2 (t) − v 2 (t) dt 7 −1 ⎧ 1

⎪ ⎪ ln t u2 (t) + v 2 (t) dt ⎪ u(x) = 22 + ln x + ⎨ 2.

0+

⎪ ⎪ ⎪ ⎩ v(x) = −14 − ln x +

1 0+

ln t u2 (t) − v 2 (t) dt

510 15 Nonlinear Fredholm Integral Equations ⎧ 1 2

⎪ 2 ⎪ ⎪ ⎨ u(x) = 22 − 11 ln x + + ln(xt) u (t) + v (t) dt 0 3. 1 ⎪

⎪ ⎪ ⎩ v(x) = −14 + 7 ln x + ln(xt) u2 (t) − v 2 (t) dt 0+

⎧ π

π2 2 ⎪ ⎪ x + u(x) = sin x + cos x − xt u2 (t) + v 2 (t) dt ⎪ ⎨ 4 0 4. π ⎪

π ⎪ 2 ⎪ ⎩ v(x) = sin x − cos x − x + xt u2 (t) − v 2 (t) dt 2 0 ⎧ π 3

2 3π 2π ⎪ ⎪ − + u (t) + v 2 (t) dt ⎪ u(x) = x + sin2 x − ⎨ 4 3 0 5. π 3 ⎪

2π ⎪ ⎪ ⎩ v(x) = x − cos2 x − π 2 x + (x − t) u2 (t) − v 2 (t) dt + 3 0 ⎧ π √

3 ⎪ ⎪ tan2 t u2 (t) + sec2 t v 2 (t) dt u(x) = sec x − 2 3 + ⎪ ⎨ 0

6.

π ⎪ √

π ⎪ 3 ⎪ ⎩ v(x) = tan x − 2 3 + + sec2 t u2 (t) − tan2 t v 2 (t) dt 3 0 1 ⎧

11 ⎪ ⎪ u(x) = x+ (x − t) v 2 (t) + w2 (t) dt ⎪ ⎪ 35 ⎪ −1 ⎪ ⎪ ⎪ 1 ⎨

20 (x − 2t) w2 (t) + u2 (t) dt x+ v(x) = x2 − 7. ⎪ 12 −1 ⎪ ⎪ ⎪ 1 ⎪ ⎪

16 ⎪ ⎪ ⎩ w(x) = x3 − x+ (x − 3t) u2 (t) + v 2 (t) dt 15 −1 π ⎧

π 4 2 ⎪ ⎪ + v (t) − w 2 (t) dt u(x) = sec x tan x − 2 + ⎪ ⎪ 4 ⎪ 0 ⎪ ⎪ ⎪ π ⎨

π 4 8. w2 (t) − u2 (t) dt v(x) = sec2 x + 1 − + ⎪ 4 ⎪ 0 ⎪ ⎪ ⎪ π ⎪

⎪ 4 ⎪ ⎩ w(x) = sec2 x + u2 (t) − v 2 (t) dt 0

15.6.2 The modiﬁed Adomian Decomposition Method The modiﬁed Adomian decomposition method [11–12] was frequently and thoroughly used in this text. The method decomposes the linear terms u(x) and v(x) by an inﬁnite sum of components of the form ∞ ∞ un (x), v(x) = vn (x), (15.294) u(x) = n=0

n=0

where the components un (x) and vn (x) will be determined recurrently. The method can be used in its standard form, or combined with the noise terms phenomenon.

15.6 Systems of Nonlinear Fredholm Integral Equations

511

However, the nonlinear functions Fi and F˜i , for i = 1, 2, in (15.269) should be replaced by the Adomian polynomials An deﬁned by n 1 dn λi ui , n = 0, 1, 2, · · · , (15.295) An = F n! dλn i=0 λ=0

Substituting the aforementioned assumptions for the linear and the nonlinear terms into the system (15.269) and using the recurrence relations we can determine the components un (x) and vn (x). Having determined these components, the series solutions and the exact solutions are readily obtained. Example 15.37 Use the modiﬁed Adomian decomposition method to solve the following system of nonlinear Fredholm integral equations π

u(x) = sin x − π + (1 + xt)u2 (t) + (1 − xt)v 2 (t) dt, 0 (15.296) π

π2 2 2 x+ v(x) = cos x + (1 − xt)u (t) − (1 + xt)v (t) dt. 2 0 Substituting the linear terms u(x) and v(x) and the nonlinear terms u2 (x) and v 2 (x) from (15.294) and (15.295) respectively into (15.296) gives ∞ un (x) = sin x − π n=0 π ∞ ∞ + (1 + xt) An (t) + (1 − xt) Bn (t) dt, ∞ n=0

0

π2 x vn (x) = cos x + 2 π

+ 0

(1 − xt)

n=0

n=0

∞

∞

An (t) − (1 + xt)

n=0

Bn (t)

dt.

n=0

(15.297) The modiﬁed decomposition method will be used here, hence we set the recursive relation u0 (x) = sin x, v0 (x) = cos x, π

(1 + xt)u20 (t) + (1 − xt)v02 (t) dt = 0, u1 (x) = −π + (15.298) 0 π

π2 (1 − xt)u20 (t) − (1 + xt)v02 (t) dt = 0. x+ v1 (x) = 2 0 This in turn gives the exact solutions (u(x), v(x)) = (sin x, cos x).

(15.299)

Example 15.38 Use the modiﬁed Adomian decomposition method to solve the following system of Fredholm integral equations

512

15 Nonlinear Fredholm Integral Equations

u(x) = x + sec2 x −

8 π3 − + 3 96

2

v(x) = x − sec x − π + 2 ln 2 +

π 4

u2 (t) + v 2 (t) dt,

0

π 4

0

u (t) − v (t) dt. 2

(15.300)

2

Proceeding as before we obtain π ∞ ∞ ∞ 4 8 π3 2 + un (x) = x + sec x − − An (t) + Bn (t) dt, 3 96 0 n=0 n=0 n=0 π4 ∞ ∞ ∞ vn (x) = x − sec2 x − π + 2 ln 2 + An (t) = Bn (t) dt. 0

n=0

n=0

n=0

(15.301) For simplicity, the modiﬁed decomposition method will be used again, therefore we set the recursive relation u0 (x) = x + sec2 x, v0 (x) = x − sec2 x, π4

2 8 π3 + u1 (x) = − − u0 (t) + v02 (t) dt = 0, (15.302) 3 96 0 π4

2 u0 (t) + v02 (t) dt = 0. v1 (x) = −π + 2 ln 2 + 0

Consequently, the other components (uj , vj ) = (0, 0), j 2. As a result, the exact solutions are given by (15.303) (u(x), v(x)) = (x + sec2 x, x − sec2 x). Example 15.39 Use the modiﬁed Adomian decomposition method to solve the following system of nonlinear Fredholm integral equations 1 xt 2

2 x sinh(x + 2) + e u (t) + e−xt v2 (t) dt, u(x) = e − x+2 0 (15.304) 1 −xt 2

2 xt 2 −x e u (t) + e v (t) dt. sinh(x − 2) + v(x) = e − x−2 0 Proceeding as before and using the modiﬁed decomposition method, we ﬁnd the recursive relation u0 (x) = ex , v0 (x) = e−x , 1 xt 2

2 e u0 (t) + e−xt v02 (t) dt = 0, (15.305) sinh(x + 2) + u1 (x) = − x+2 0 1 −xt 2

2 e u0 (t) + ext v02 (t) dt = 0. sinh(x − 2) + v1 (x) = − x−2 0 The exact solutions are given by (15.306) (u(x), v(x)) = (ex , e−x ).

15.6 Systems of Nonlinear Fredholm Integral Equations

513

Example 15.40 Use the modiﬁed Adomian decomposition method to solve the following system of nonlinear Fredholm integral equations 1 x−3t 2

x x+1 u(x) = e − e e + v (t) + ex−6t w2 (t) dt, v(x) = e2x − 2ex + w(x) = e

3x

− 2e +

0 1

ex−6t w2 (t) + ex−2t u2 (t) dt,

(15.307)

0 1

x

ex−2t u2 (t) + ex−4t v 2 (t) dt.

0

Using the modiﬁed decomposition method, we set the recurrence relation u0 (x) = ex , v0 (x) = e2x , w0 (x) = e3x ,

u1 (x) = −ex+1 + v1 (x) = −2ex +

ex−3t v02 (t) + ex−6t w02 (t) dt = 0,

ex−6t w02 (t) + ex−2t u20 (t) dt = 0,

0

w1 (x) = −2ex +

0 1

1

1

(15.308)

ex−2t u2 (t) + ex−4t v 2 (t) dt = 0.

0

Consequently, the exact solutions are given by (u(x), v(x), w(x)) = (ex , e2x , e3x ).

(15.309)

Exercises 15.6.2 Use the modiﬁed Adomian decomposition method to solve the following systems of nonlinear Fredholm integral equations 1 ⎧

4 ⎪ ⎪ u(x) = x − + t u2 (t) + v 2 (t) dt ⎪ ⎨ 7 −1 1. 1 ⎪

4 ⎪ ⎪ t u2 (t) − v 2 (t) dt ⎩ v(x) = x2 + x3 + + 7 −1 ⎧ π

2 ⎪ ⎪ u (t) + v 2 (t) dt u(x) = sin x + cos x − 2π + ⎪ ⎨ 0 2. π ⎪

⎪ ⎪ xt u2 (t) − v 2 (t) dt ⎩ v(x) = sin x − cos x − πx + 0

⎧ ⎪ ⎪ ⎪ ⎨ u(x) = x + sin x + (4 − 2π) + 3.

π

0

2 u (t) − v 2 (t) dt

⎪ ⎪ ⎪ ⎩ v(x) = x − cos x + (8 − 4π − 2π 2 ) +

π

0

t u2 (t) − v 2 (t) dt

514

15 Nonlinear Fredholm Integral Equations

⎧ π

π−8 4 2 ⎪ ⎪ + u (t) + v 2 (t) dt ⎪ u(x) = sec x + ⎨ 4 0 4. π ⎪

π ⎪ 4 ⎪ ⎩ v(x) = tan x − x + x u2 (t) − v 2 (t) dt 4 0 ⎧ π

π 4 2 ⎪ ⎪ + u (t) − v 2 (t) dt u(x) = sec x − ⎪ ⎨ 4 0 5. π ⎪

π ⎪ 4 ⎪ ⎩ v(x) = tan x − x + x u2 (t) − v 2 (t) dt 4 0 √ ⎧ π

3 6 2 ⎪ ⎪ + u (t) − v 2 (t) dt u(x) = sec x tan x + ⎪ ⎨ 3 0 6. √ π ⎪

⎪ 3 11 6 ⎪ ⎩ v(x) = sec2 x − x u2 (t) + v 2 (t) dt x+ 27 0 ⎧ π 2

π + 4 4 ⎪ ⎪ u (t) + u(t)v(t) dt ⎪ ⎨ u(x) = sec x − 4 + 0 7. π ⎪

π − 4 ⎪ 4 2 ⎪ ⎩ v(x) = cos x + + u (t) − u(t)v(t) dt 4 0 ⎧ π

π−5 4 2 ⎪ ⎪ + u (t) + u(t)v 2 (t) dt u(x) = tan x + ⎪ ⎨ 4 0 8. π ⎪

π−3 ⎪ 4 2 ⎪ ⎩ v(x) = cos x + + u (t) − u(t)v 2 (t) dt 4 0 π ⎧

π − 10 4 2 ⎪ ⎪ u(x) = sec x + + v (t) + w 2 (t) dt ⎪ ⎪ 8 ⎪ 0 ⎪ ⎪ ⎪ π ⎨

π + 10 4 9. + w 2 (t) + u2 (t) dt v(x) = tan x − ⎪ 8 ⎪ 0 ⎪ ⎪ ⎪ π ⎪

⎪ π−8 4 2 ⎪ ⎩ w(x) = cos x + + u (t) + v 2 (t) dt 4 0 π ⎧

3π − 10π 2 4 2 ⎪ 2x+ ⎪ + v (t) + w 2 (t) dt u(x) = 1 + π sec ⎪ ⎪ 6 ⎪ 0 ⎪ ⎪ ⎪ π ⎨ 2

21π + 10π 4 10. v(x) = 1 − π sec2 x − + w2 (t) + u2 (t) dt ⎪ 6 ⎪ 0 ⎪ ⎪ ⎪ π ⎪ 2

⎪ π 3π + 16π 4 2 ⎪ ⎩ w(x) = 1 + sec2 x − + u (t) + v 2 (t) dt 2 6 0 π ⎧ 1 3 ⎪ ⎪ u(x) = sec x − + (v(t)w(t)) dt ⎪ ⎪ 2 ⎪ 0 ⎪ ⎪ ⎪ π ⎨ π 3 11. (w(t)u(t)) dt v(x) = tan x + + ⎪ 3 ⎪ 0 ⎪ ⎪ ⎪ π ⎪ ⎪ ⎪ ⎩ w(x) = cos x − 1 + 3 (u(t)v(t)) dt 0

References

515

π ⎧ 2−π 4 ⎪ ⎪ (v(t)w(t)) dt + u(x) = sec2 x + ⎪ ⎪ 8 ⎪ 0 ⎪ ⎪ ⎪ π ⎨ 1 4 12. (w(t)u(t)) dt v(x) = − sec2 x − + ⎪ 3 ⎪ 0 ⎪ ⎪ ⎪ π ⎪ ⎪ π ⎪ ⎩ w(x) = tan2 x − + 4 (u(t)v(t)) dt 4 0

References 1. H.T. Davis, Introduction to Nonlinear Diﬀerential and Integral Equations, Dover Publications, New York, (1962). 2. A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 3. L.M. Delves and J. Walsh, Numerical Solution of Integral Equations,Oxford University Press, London, (1974). 4. R. Kress, Linear Integral Equations, Springer, Berlin, (1999). 5. J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Diﬀerential Equations, Yale University Press, New Haven, (1923). 6. J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg., 178 (1999) 257–262. 7. D.L. Phillips, A technique for the numerical solution of certain integral equations of the ﬁrst kind, J. Asso. Comput. Mach, 9 (1962) 84–96. 8. A.N. Tikhonov, On the solution of incorrectly posed problem and the method of regularization, Soviet Math, 4 (1963) 1035–1038. 9. R.F. Churchhouse, Handbook of Applicable Mathematics, Wiley, New York, (1981). 10. Y. Cherruault and V. Seng, The resolution of non-linear integral equations of the ﬁrst kind using the decomposition method of Adomian, Kybernetes, 26 (1997) 109–206. 11. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 12. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997).

Chapter 16

Nonlinear Fredholm Integro-Diﬀerential Equations

16.1 Introduction The linear Fredholm integral equations and the linear Fredholm integrodiﬀerential equations were presented in Chapters 4 and 6 respectively. In Chapter 15, the nonlinear Fredholm integral equations were examined. It is our goal in this chapter to study the nonlinear Fredholm integro-diﬀerential equations [1–7] and the systems of nonlinear Fredholm integro-diﬀerential equations. The nonlinear Fredholm integro-diﬀerential equations of the second kind is of the form 1 K(x, t)F (u(t)) dt, (16.1) u(n) (x) = f (x) + λ 0 (n)

where u (x) is the n th derivative of u(x). The kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x). In this chapter, we will mostly use degenerate or separable kernels. A degenerate or a separable kernel is a function that can be expressed as the sum of product of two functions each depends only on one variable. Such a kernel can be expressed in the form n gi (x) hi (t). (16.2) K(x, t) = i=1

Several analytic and numerical methods have been used to handle the nonlinear Fredholm integro-diﬀerential equations. In this text we will apply three of the methods used in this text, namely, the direct computation method, the variational iteration method (VIM), and the Taylor series solution method to handle the nonlinear Fredholm integro-diﬀerential equations. The emphasis in this text will be on the use of these methods rather than proving theoretical concepts of convergence and existence. The theorems of uniqueness, existence, and convergence are important and can be found in the literature. A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

518

16 Nonlinear Fredholm Integro-Diﬀerential Equations

The concern will be on the determination of the solutions u(x) of nonlinear Fredholm integro-diﬀerential equations of the second kind.

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations The linear Fredholm integro-diﬀerential equation, where both diﬀerential and integral operators appear together in the same equation, has been studied in Chapter 6. In this section, the nonlinear Fredholm integro-diﬀerential equation will be examined. The standard form of the nonlinear Fredholm integrodiﬀerential equations of reads 1 K(x, t)F (u(t))dt, (16.3) u(i) (x) = f (x) + λ 0 (i)

di u dxi ,

and F (u(x)) is a nonlinear function of u(x) such as where u (x) = u2 (x), sin(u(x)), and eu(x) . Because the equation in (16.3) combines the differential operator and the integral operator, then it is necessary to deﬁne initial conditions u(0), u (0), . . . , u(i−1) (0) for the determination of the particular solution u(x) of this equation. In Chapter 6, we applied four methods to handle the linear Fredholm integro-diﬀerential equations of the second kind. In this section we will use only three of the methods that we used in Chapter 6. However, the other methods presented in Chapter 6 can be used as well. In what follows we will apply the direct computation method, the variational iteration method (VIM), and the series solution method to handle nonlinear Fredholm integro-diﬀerential equations of the second kind (16.3).

16.2.1 The Direct Computation Method The direct computation method has been extensively introduced in this text. Without loss of generality, we may assume a standard form to the Fredholm integro-diﬀerential equation given by b (n) u (x) = f (x) + K(x, t) F (u(t))dt, u(k) (0) = bk , 0 k (n − 1), (16.4) a

where u(n) (x) indicates the n th derivative of u(x) with respect to x, F (u(t)) is a nonlinear function of u(x), and bk are the initial conditions. It is important to point out that this method will be applied for equations where the kernels are degenerate or separable of the form n K(x, t) = gk (x)hk (t). (16.5) k=1

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations

519

Substituting (16.5) into the nonlinear Fredholm integro-diﬀerential equation (16.4) leads to b u(n) (x) = f (x) + g1 (x) h1 (t) F (u(t))dt a b b h2 (t) F (u(t))dt + · · · + gn (x) hn (t) F (u(t))dt. +g2 (x) a

a

(16.6) Each integral at the right side of (16.6) depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (16.6) becomes u(n) (x) = f (x) + α1 g1 (x) + α2 g2 (x) + · · · + αn gn (x), (16.7) where

b

hi (t) F (u(t))dt, 1 i n.

αi =

(16.8)

a

Integrating both sides of (16.7) n times, using the initial conditions, we obtain 1 1 xn−1 u(n−1) u(x) = u(0) + xu (0) + x2 u (0) + · · · + 2! (n − 1)! (16.9) +L−1 (f (x) + α1 g1 (x) + α2 g2 (x) + · · · + αn gn (x)) , −1

where L is the n-fold integral operator. Substituting (16.9) into (16.8) gives a system of n algebraic equations that can be solved to determine the constants αi , 1 i n. Using the obtained numerical values of αi into (16.7), the solution u(x) of the nonlinear Fredholm integro-diﬀerential equation (16.4) is readily obtained. It is interesting to point out that we may get more than one value for one or more of αi , 1 i n. This is normal because the equation is nonlinear and the solution u(x) may not be unique for nonlinear problems. In what follows we present some examples to illustrate the use of the direct computation method. Example 16.1 Solve the nonlinear Fredholm integro-diﬀerential equation by using the direct computation method π π2 xtu2 (t) dt, u(0) = 0. (16.10) u (x) = cos x − x+ 4 0 This equation may be written as π2 u (x) = cos x + α − x, u(0) = 0, (16.11) 4 where π tu2 (t) dt.

α=

(16.12)

0

Integrating both sides of (16.11) from 0 to x, and using the initial condition, we ﬁnd

520

16 Nonlinear Fredholm Integro-Diﬀerential Equations

u(x) = sin x +

α π2 − 2 8

x2 .

(16.13)

Substituting (16.13) into (16.12), evaluating the resulting integral, and solving the resulting equation for α we obtain π2 π 8 − 96π 3 + 576π + 96 , . (16.14) α= 4 4π 6 Consequently, the exact solutions are given by 12 72 12 u(x) = sin x, sin x − + 5 + 6 x2 . (16.15) π3 π π Example 16.2 Solve the nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 1 1 − e2 x+ xu2 (t) dt, u(0) = 1. (16.16) u (x) = ex + 2 0 This equation may be written as 1 − e2 + 2α x, u(0) = 1, (16.17) u (x) = ex + 2 where 1 α= u2 (t) dt. (16.18) 0

Integrating both sides of (16.17) from 0 to x, and by using the initial conditions we obtain 1 − e2 + 2α 2 x . u(x) = ex + (16.19) 4 Substituting (16.19) into (16.18) and proceeding as before we get e2 − 1 e2 − 40e + 119 , . (16.20) α= 2 2 The exact solutions are therefore given by u(x) = ex , ex + (30 − 10e)x2 .

(16.21)

Example 16.3 Solve the nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 19 11 1 1 u (x) = 2+ x+ x2 + (xt+x2 t2 ) (u(t)−u2 (t)) dt, u(0) = 1, u (0) = 1. 15 35 2 −1 (16.22) This equation may be written as 11 1 19 1 α+ x+ β+ x2 , u(0) = 1, u (0) = 1, (16.23) u (x) = 2 + 2 15 2 35 obtained by setting

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations

1

α= −1

t(u(t) − u2 (t)) dt,

1

β= −1

521

t2 (u(t) − u2 (t)) dt.

(16.24)

Integrating both sides of (16.23) two times from 0 to x, and by using the given initial conditions we ﬁnd 1 1 11 19 2 3 (16.25) α+ x + β+ x4 . u(x) = 1 + x + x + 12 90 24 420 Substituting (16.25) into (16.24), evaluating the integrals, and solving the resulting equations we ﬁnd 38 22 (16.26) α=− , β=− . 15 35 The exact solution is given by u(x) = 1 + x + x2 .

(16.27)

Example 16.4 Solve the nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 2 1 π cos(x − t) u2 (t) dt, u (x) = sin x + 3 2 0 (16.28) u(0) = 1, u (0) = 0, u (0) = −1. This equation may be written as 2 1 1 u (x) = sin x + α cos x + β sin x, 3 2 2 (16.29) u(0) = 1, u (0) = 0, u (0) = −1, obtained by setting π π 2 cos tu (t) dt, β = sin tu2 (t) dt. (16.30) α= 0

0

Integrating both sides of (16.29) three times from given initial conditions we obtain 2 1 1 1 u(x) = + β cos x + α(x − sin x) + β− 3 2 2 4 Proceeding as in the previous examples gives 2 α = 0, β = . 3 The exact solution is therefore given by u(x) = cos x.

0 to x, and by using the 1 6

1 1 x2 − β + . (16.31) 2 3 (16.32) (16.33)

Exercises 16.2.1 Solve the following nonlinear Fredholm integro-diﬀerential equations by using the direct computation method

522 1. u (x) = 1 −

16 Nonlinear Fredholm Integro-Diﬀerential Equations 1 17 x+ 24 2

1 0

xtu2 (t) dt, u(0) = 1

149 2 1 1 2 2 x tu (t) dt, u(0) = 1 x + 240 4 0 π 1 2 3. u (x) = cos x − x + xtu2 (t) dt, u(0) = 1 4 −π 2 1 2 1 π (π + 8)x + 4. u (x) = sin x + xt(u(t) − u2 (t)) dt, u(0) = 0 16 4 0 1 e e2 1 1 x(u(t) + u2 (t)) dt, u(0) = 1 + + x+ 5. u (x) = 1 + ex − 3 4 8 4 0 1 1 1 xt(1 + u(t) − u2 (t)) dt, u(0) = 2 6. u (x) = ex + (3 + e2 )x + 16 4 0 π2 1 π π x+ + 7. u (x) = − sin x − (x − t)u2 (t) dt, u(0) = 0, u (0) = 1 16 32 8 0 7 π 1 π sin(x − t)u2 (t) dt, u(0) = 0, u (0) = 0 8. u (x) = cos x + sin x + 3 2 2 0 1 1 1 x−2t 2 9. u (x) = ex + e u (t) dt, u(0) = 1, u (0) = 1 2 2 0 3 1 1 x−3t 3 10. u (x) = ex + e u (t) dt, u(0) = u (0) = u (0) = 1 4 4 0 1 1 x 1 11. u (x) = 8e2x − ex−4t u2 (t) dt, u(0) = 1, u (0) = 2, u (0) = 4 e + 16 16 0 π 1 π (u(t) − u2 (t)) dt, u(0) = −u (0) = 1, u (0) = 0 12. u (x) = sin x + + 16 8 0 2. u (x) = 1 + 2x −

16.2.2 The Variational Iteration Method The variational iteration method [8–10] was used before in previous chapters. The method handles linear and nonlinear problems in a straightforward manner. Unlike the Adomian decomposition method where we determine distinct components of the exact solution, the variational iteration method gives rapidly convergent successive approximations of the exact solution if such a closed form solution exists. The standard ith order nonlinear Fredholm integro-diﬀerential equation is of the form 1

u(i) (x) = f (x) +

K(x, t)F (u(t))dt,

(16.34)

0 i

where u(i) (x) = ddxui , and F (u(x)) is a nonlinear function of u(x). The initial conditions should be prescribed for the complete determination of the exact solution.

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations

523

The correction functional for the nonlinear integro-diﬀerential equation (16.34) is 1 t un+1 (x) = un (x) + λ(t) u(i) (t) − f (t) − K(t, r)F (˜ u (r)) dr dt. n n 0

0

(16.35) To apply this method in an eﬀective way, we should follow two essential steps: (i) It is required ﬁrst to determine the Lagrange multiplier λ that can be identiﬁed optimally via integration by parts and by using a restricted variation. The Lagrange multiplier λ may be a constant or a function. (ii) Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), n 0 of the solution u(x). The zeroth approximation u0 can be any selective function. However, the initial values are preferably used for the selective zeroth approximation u0 . Consequently, the solution is given by u(x) = lim un(x). n→∞

(16.36)

The VIM will be illustrated by studying the following examples. Example 16.5 Use the variational iteration method to solve the nonlinear Fredholm integrodiﬀerential equation π 1 π u (x) = cos x − x + xu2 (t) dt, u(0) = 0. (16.37) 48 24 0 The correction functional for this equation is given by π x 1 π un (t) − cos t + t − tu2n (r)dr dt, (16.38) un+1 (x) = un (x) − 48 24 0 0 where we used λ = −1 for ﬁrst-order integro-diﬀerential equation. We can use the initial condition to select u0 (x) = u(0) = 0. Using this selection into the correction functional gives the following successive approximations u0 (x) = 0, u1 (x) = sin x − 0.03272492349x2, u2 (x) = sin x − 0.00663791983x2, (16.39) 2 u3 (x) = sin x − 0.00156723251x , u4 (x) = sin x − 0.00038016125x2, and so on. Using (16.36) gives the exact solution by u(x) = sin x.

(16.40)

Example 16.6 Use the variational iteration method to solve the nonlinear Fredholm integrodiﬀerential equation

524

16 Nonlinear Fredholm Integro-Diﬀerential Equations

π 1 π(π 2 + 3) x+ xtu2 (t) dt, u(0) = 0. (16.41) 512 64π 0 The correction functional for this equation is given by un+1 (x) = un(x) π x 1 π(π 2 + 3) un (t) − cos t + t sin t + t− − tru2n (r) dr dt. 512 64π 0 0 (16.42) We next select u0 (x) = 0 to ﬁnd the successive approximations u0 (x) = 0, u1 (x) = x cos x − 0.03948345181x2, u2 (x) = x cos x + 0.01017026496x2, (16.43) 2 u3 (x) = x cos x − 0.002418466659x , u4 (x) = x cos x + 0.000587237401x2, and so on. Proceeding as before, the exact solution is given by u(x) = x cos x. (16.44) u (x) = cos x − x sin x −

Example 16.7 Use the variational iteration method to solve the nonlinear Fredholm integrodiﬀerential equation π 1 3π x+ xu2 (t)dt, u(0) = 2, u (0) = 0. (16.45) u (x) = − cos x − 128 64 0 The correction functional for this equation is given by π x 1 3π t− (t − x) un (t) + cos t + tu2n (r) dr dt. un+1 (x) = un (x) + 128 64 0 0 (16.46) We can use the initial condition to select u0 (x) = u(0) + xu (0) = 2. Using this selection into the correction functional gives the following successive approximations u0 (x) = 2, u1 (x) = 1 + cos x + 0.02045307718x3, u2 (x) = 1 + cos x + 0.00118839931x3, (16.47) u3 (x) = 1 + cos x + 0.00004332607x3, u4 (x) = 1 + cos x + 0.00000152382x3, and so on. The VIM gives the exact solution by u(x) = 1 + cos x. (16.48) Example 16.8 Use the variational iteration method to solve the nonlinear Fredholm integrodiﬀerential equation

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations

u (x) = xex +

1079 x 1 e + 360 120

1

525

ex−2t u2 (t) dt, u(0) = 0, u (0) = 1, u (0) = 2.

0

(16.49) The correction functional for this equation is given by un+1 (x) = un (x) 1 1 1079 t 1 1 t t−2r 2 (t − x)2 u e (t) − te − − e u (r) dr dt. − n n 360 120 0 0 2 (16.50) We can use the initial condition to select u0 (x) = x + x2 . Using this selection into the correction functional gives the following successive approximations u0 (x) = x + x2 , u1 (x) = xex + (0.88212x + 0.441063x2 − 0.88212ex + 0.88212) × 10−3 , u2 (x) = xex + (0.267448x + 0.1337x2 − 0.26744ex + 0.26744) × 10−6 , u3 (x) = xex + (0.8115x + 0.40602x2 − 0.81286ex + 0.81135) × 10−10 , (16.51) and so on. The VIM admits the use of u(x) = lim un(x), (16.52) n→∞

that gives the exact solution by u(x) = xex .

(16.53)

Exercises 16.2.2 Use the variational iteration method to solve the nonlinear Fredholm integro-diﬀerential equations π π 1 1. u (x) = sin x − xu2 (t) dt, u(0) = 0 + 80 120 0 π 1 1 π 2. u (x) = cos x − x− + xu2 (t) dt, u(0) = 1 80 30 120 0 π π2 1 xtu2 (t) dt, u(0) = 0 (3 − π 2 )x + 3. u (x) = x + x cos x + 192 24 0 1 1 x 1 4. u (x) = 2e2x − ex−4t u2 (t) dt, u(0) = 1 e + 24 24 0 1 1 1 5. u (x) = −2e−2x − xex+4t u2 (t) dt, u(0) = 1 xex + 24 24 0 π 1 π 6. u (x) = cos x + sin x − + xu2 (t) dt, u(0) = −1 96 96 0 π π π2 1 (x − t)u2 (t) dt, u(0) = 0, u (0) = 1 x+ + 7. u (x) = − sin x − 36 72 18 0

526

16 Nonlinear Fredholm Integro-Diﬀerential Equations

8. u (x) = − cos x −

π2 1 x+ 288 72

π 0

xtu2 (t) dt, u(0) = 1, u (0) = 0

1 2 1 1 x(t − u2 (t)) dt, u(0) = 1, u (0) = 1 (e − 2)x + 4 2 0 π π 1 x(u(t) − u2 (t)) dt, u(0) = 1, u (0) = 0, u (0) = −1 10. u (x) = sin x + x+ 144 72 0 π 1 π 11. u (x) = sin x − cos x − x+ xu2 (t) dt, u(0) = 1, u (0) = 1, u (0) = −1 100 100 0 1 1 1 (e2 − 3)x + 12. u (x) = ex + x(2u(t) − u2 (t)) dt, u(0) = 2, u (0) = 200 100 0 1, u (0) = 1 9. u (x) = ex +

16.2.3 The Series Solution Method The series solution method depends mainly on the Taylor series for analytic functions [1]. A real function u(x) is called analytic if it has derivatives of all orders such that the generic form of Taylor series at x = 0 can be written as ∞ u(x) = a n xn . (16.54) n=0

The Taylor series method, or simply the series solution method will be used in this section for solving nonlinear Fredholm integro-diﬀerential equations of the second kind. We will assume that the solution u(x) of the nonlinear Fredholm integro-diﬀerential equation 1 (n) u (x) = f (x) + λ K(x, t)F (u(t))dt, u(k) (0) = k!ak , 0 k (n − 1), 0

(16.55) is analytic, and therefore possesses a Taylor series of the form given in (16.54), where the coeﬃcients an will be determined recurrently. The ﬁrst few coeﬃcients ak can be determined by using the initial conditions so that 1 1 (16.56) a0 = u(0), a1 = u (0), a2 = u (0), a3 = u (0), 2! 3! and so on. The remaining coeﬃcients ak of (16.54) will be determined by applying the series solution method to the nonlinear Fredholm integrodiﬀerential equation (16.55). Substituting (16.54) into both sides of (16.55) gives (n) ∞ ∞ 1 k k (16.57) ak x = T (f (x)) + K(x, t)F ak t dt, k=0

0

k=0

where T (f (x)) is the Taylor series for f (x). The integro-diﬀerential equation (16.55) will be converted to a traditional integral in (16.57) where instead of

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations

527

integrating the unknown function F (u(x)), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integral in (16.57), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0, where some of these coeﬃcients will be used from the initial conditions. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (16.54). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then a truncated series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. The following examples will be used to illustrate the series solution method. Example 16.9 Solve the nonlinear Fredholm integro-diﬀerential equation by using the series solution method 1 226 2 2 x + u (x) = 1 − x − (xt + x2 t2 )u2 (t))dt, u(0) = 1. (16.58) 15 105 −1 Substituting u(x) by the series ∞ an xn , (16.59) u(x) = n=0

into both sides of the equation (16.58) leads to ∞ 1 ∞ 226 2 2 n 2 2 n 2 x + an x = 1− x− an t ) dt. (16.60) (xt + x t )( 15 105 −1 n=0 n=0 Evaluating the integral at the right side, using a0 = 1, we ﬁnd a1 + 2a2 x + 3a3 x2 + 4a4 x3 + 5a5 x4 + · · · 4 4 4 4 4 2 4 = 1 + − + a1 + a3 a4 + a1 a2 + a1 a4 + a2 a3 + a3 x 15 3 9 5 7 7 5 4 2 2 4 4 2 2 2 2 52 2 2 4 − + a3 + a2 a4 + a1 a3 + a2 + a4 + a2 + a1 + a4 x2 . 35 9 9 7 7 7 5 5 11 (16.61) Equating the coeﬃcients of like powers of x in both sides, and solving the system of equations we obtain two sets of solutions a0 = 1, a1 = 1, a2 = 1, ar = 0, r 3, (16.62) Consequently, the exact solutions is given by u(x) = 1 + x + x2 ,

(16.63)

528

16 Nonlinear Fredholm Integro-Diﬀerential Equations

Example 16.10 Solve the nonlinear Fredholm integro-diﬀerential equation by using the series solution method 1 1 1 2 u (x) = ex − (e2 − 1)x + xu (t)dt, u(0) = 1. (16.64) 4 2 0 Substituting u(x) by the series ∞ u(x) = an xn , (16.65) n=0

into both sides of the equation (16.64) leads to ∞ 1 ∞ 1 2 n x n 2 an x = T1 (e − (e − 1)x) + (an t ) dt, x 4 0 n=0 n=0

(16.66)

where T1 is the Taylor series about x = 0. Evaluating the integral at the right side, using a0 = 1, and proceeding as before we ﬁnd 1 an = , n 0. (16.67) n! Using (16.65) gives the exact solution (16.68) u(x) = ex . It is worth noting that we used the series assumption up to O(x12 ) to get this result. Example 16.11 Solve the nonlinear Fredholm integro-diﬀerential equation by using the series solution method π 1 π π2 1 π + x+ + + u (x) = cos x − (x − t)u2 (t)dt, u(0) = 1. 8 3 6 16 12 0 (16.69) Substituting u(x) by the series ∞ an xn , (16.70) u(x) = n=0

into both sides of the equation (16.69) leads to ∞ π 1 π π2 n an x = T1 cos x − + x+ + 8 3 6 16 n=0 ⎛ ⎞ 2 π ∞ 1 ⎝(x − t) an tn ⎠ dt. + 12 0 n=0

(16.71)

Evaluating the integral at the right side, using a0 = 1, and proceeding as before we ﬁnd

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations

529

a0 = 1, a2n+1 =

∞ n=0

(−1)n , n 0, (2n + 1)!

(16.72)

a2n = 0, n 0. Consequently, the series solution is given by ∞ (−1)n 2n+1 u(x) = 1 + x , (2n + 1)! n=0 that converges to the exact solution u(x) = 1 + sin x.

(16.73)

(16.74)

Example 16.12 Solve the nonlinear Fredholm integro-diﬀerential equation by using the series solution method 146 1 1 xtu2 (t)dt, u(0) = 1, u (0) = 0. (16.75) u (x) = 10 − x+ 35 2 −1 Substituting u(x) by the series ∞ u(x) = an xn , (16.76) n=0

into both sides of the equation (16.75), evaluating the integral at the right side, using a0 = a2 = 0, a1 = 1, and proceeding as before we ﬁnd a0 = 1, a1 = 0, a2 = 5, a3 = −1, ak = 0, k 4. Consequently, the exact solution is given by u(x) = 1 + 5x2 − x3 .

(16.77) (16.78)

Exercises 16.2.3 Solve the following nonlinear Fredholm integro-diﬀerential equations by using the series solution method 1 193 1 47 1. u (x) = − − x+ (x − t)u2 (t) dt, u(0) = 1 45 90 12 −1 1 53 1 197 (x − t)u2 (t) dt, u(0) = 1 − x + 3x2 + 2. u (x) = 45 630 12 −1 1 1 82 x+ 3. u (x) = 1 + xtu2 (t) dt, u(0) = 1 45 12 −1 1 1 2 1 xt(u2 (t) − u(t)) dt, u(0) = 1 (e − 3)x + 4. u (x) = ex − 96 24 0 1 1 x−1 1 5. u (x) = ex − + ex−2t (u2 (t) − u(t)) dt, u(0) = 1 e 24 24 0

530 6. u (x) = −2 −

16 Nonlinear Fredholm Integro-Diﬀerential Equations 1 4 x+ 15 2

1 −1

xtu2 (t) dt, u(0) = u (0) = 1

1 1 2 1 xtu2 (t) dt, u(0) = u (0) = 1 (e − 1)x + 48 12 0 1796 1 1 2 2 8. u (x) = xt u (t) dt, u(0) = 0, u (0) = 1 x+ 315 2 0

7. u (x) = ex −

16.3 Homogeneous Nonlinear Fredholm Integro-Diﬀerential Equations Substituting f (x) = 0 into the nonlinear Fredholm integral equation of the second kind b (i) K(x, t)F (u(t))dt, (16.79) u (x) = f (x) + λ a

gives the homogeneous nonlinear Fredholm integral equation of the second kind given by b u(i) (x) = λ K(x, t)F (u(t))dt, (16.80) a

where F (u(t)) is a nonlinear function of u(t). The initial conditions should be prescribed to determine the exact solution. In this section we will focus our study on the homogeneous nonlinear Fredholm integral equation (16.80) for the speciﬁc case where the kernel K(x, t) is separable. The aim for studying the homogeneous nonlinear Fredholm equation is to ﬁnd nontrivial solution. Moreover, the Adomian decomposition method is not applicable here because it depends mainly on assigning a nonzero value for the zeroth component u0 (x), and f (x) = 0 in this kind of equations. The direct computation method works eﬀectively to handle the homogeneous nonlinear Fredholm integro-diﬀerential equations.

16.3.1 The Direct Computation Method The direct computation method was used before in this chapter. This method replaces the homogeneous nonlinear Fredholm integro-diﬀerential equations by a single algebraic equation or by a system of simultaneous algebraic equations depending on the number of terms of the separable kernel K(x, t). The direct computation method handles Fredholm integro-diﬀerential equations, homogeneous or nonhomogeneous, in a direct manner and gives the solution in an exact form and not in a series form. As stated before, the direct computation method will be applied to equations where the kernel K(x, t0 ) is degenerate or separable kernel of the form

16.3 Homogeneous Nonlinear Fredholm Integro-Diﬀerential Equations

K(x, t) =

n

gk (x)hk (t).

531

(16.81)

k=1

The direct computation method can be applied as follows: 1. We ﬁrst substitute (16.81) into the homogeneous nonlinear Fredholm integro-diﬀerential equation b u(i) (x) = λ K(x, t) F (u(t))dt. (16.82) a

2. This substitution leads to b h1 (t) F (u(t))dt + λg2 (x) u(i) (x) = λg1 (x) a

b

h2 (t) F (u(t))dt + · · · a

b

+λgn (x)

hn (t) F (u(t))dt.

(16.83)

a

3. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Consequently, Equation (16.83) becomes u(i) (x) = λα1 g1 (x) + λα2 g2 (x) + · · · + λαn gn(x), where

b

hi (t) F (u(t))dt, 1 i n.

αi =

(16.84) (16.85)

a

4. Integrating both sides of (16.84) i times from 0 to x, and using the given initial conditions we obtain an expression for u(x) in terms of αi and x. 5. Substituting the resulting expression for u(x) into (16.85) gives a system of n simultaneous algebraic equations that can be solved to determine the constants αi , 1 i n. Using the obtained values of αi into (16.84), the solution u(x) of the integro-diﬀerential equation (16.80) follows immediately. Example 16.13 Solve the homogeneous nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 1 1 xu2 (t) dt, u(0) = 1. (16.86) u (x) = λ 12 0 This equation may be written as 1 u (x) = λαx, u(0) = 1, (16.87) 12 where 1 α= u2 (t) dt. (16.88) 0

Integrating both sides of (16.87) from 0 to x, and using the initial condition, we ﬁnd 1 (16.89) u(x) = 1 + λαx2 . 24

532

16 Nonlinear Fredholm Integro-Diﬀerential Equations

Substituting (16.89) into (16.88), evaluating the integral, and solving the resulting equation for α we obtain √ 2880 − 80λ ± 32 8100 − 450λ − 5λ2 α= . (16.90) 2λ2 This in turn gives the exact solutions √ 2880 − 80λ ± 32 8100 − 450λ − 5λ2 2 (16.91) x . u(x) = 1 + 48λ We now consider the following three cases: 1. Using λ = 0 into (16.89) gives the exact solution u(x) = 1. However, u(x) is undeﬁned by using λ = 0 into (16.91). Hence, λ = 0 is a singular point of homogeneous nonlinear √ Fredholm integro-diﬀerential equation (16.86). 2. For λ = −45 ± 27 5, Equation (16.91) gives the exact solutions √ u(x) = 1 ± 5 x2 . (16.92) √ Consequently, there are two bifurcation√points, namely −45 √ ± 27 5 for this equation. This shows that for −45 − 27 5 < λ < −45 + 27 5, then equation √ (16.86) gives two√real solutions, but has no real solutions for λ > −45 + 27 5 or λ < −45 − 27 5. √ √ 3. For −45 − 27 5 < λ < −45 + 27 5, Equation (16.86) gives two exact real solutions. This is normal for nonlinear problems, where solution may not be unique. Example 16.14 Solve the homogeneous nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 1 u (x) = λ ex−t u2 (t) dt, u(0) = 0. (16.93) 0

This equation can be rewritten as u (x) = αλex , u(0) = 0, where

1

α=

e−t u2 (t) dt.

(16.94) (16.95)

0

Integrating both sides of (16.94) from 0 to x, and using the initial condition, we ﬁnd u(x) = λα(ex − 1). (16.96) Substituting (16.96) into (16.95), evaluating the integral, and solving the resulting equation for α we obtain 1 . (16.97) α= 2 λ (2 sinh 1 − 2) This in turn gives the exact solution by ex − 1 . (16.98) u(x) = λ(2 sinh 1 − 2)

16.3 Homogeneous Nonlinear Fredholm Integro-Diﬀerential Equations

533

Notice that λ = 0 is a singular point of this equation. Example 16.15 Solve the homogeneous nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 1 u (x) = λ (xt + x2 t2 ) u2 (t) dt, u(0) = 0. (16.99) −1

This equation can be rewritten as u (x) = αλx + βλx2 , u(0) = 0, where

1

2

t u (t) dt,

α=

1

β=

−1

t2 u2 (t) dt.

(16.100) (16.101)

−1

Integrating both sides of (16.100) from 0 to x, and using the initial condition, we ﬁnd 1 1 u(x) = λαx2 + λβx3 . (16.102) 2 3 Substituting (16.102) into (16.101), evaluating the integrals, and solving the resulting system of equations for α and β we obtain √ 14 5 81 21 α = 0, , β = 2, 2. (16.103) 3λ2 2λ 2λ This in turn gives the exact solutions √ 27 3 7 5 2 7 3 x , x + x . u(x) = (16.104) 2λ 3λ 2λ Notice that λ = 0 is a singular point of this equation. Example 16.16 Solve the homogeneous nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 1 (1 + xt) (u(t) − u2 (t)) dt, u(0) = u (0) = 0. (16.105) u (x) = λ −1

This equation can be rewritten as u (x) = αλ + βλx, u(0) = u (0) = 0, where 1 1 α= (u(t) − u2 (t)) dt, β = t (u(t) − u2 (t)) dt. −1

(16.106)

(16.107)

−1

Integrating both sides of (16.106) twice from 0 to x, and using the initial conditions, we ﬁnd 1 1 (16.108) u(x) = λαx2 + λβx3 . 2 6 Substituting (16.108) into (16.107), evaluating the integrals, and solving the resulting system of equations for α and β we obtain

534

16 Nonlinear Fredholm Integro-Diﬀerential Equations

7 15(29λ + 165)(λ − 15) 10(λ − 3) 7(λ − 15) , , β = 0, ± . (16.109) 3λ2 5λ2 25λ2 (i) For α = 10(λ−3) 3λ2 , β = 0, the exact solution is given by 5(λ − 3) 2 x . u(x) = (16.110) 3λ √

α=

7

15(29λ+165)(λ−15)

,β = ± , the exact solutions are (ii) For α = 7(λ−15) 5λ2 25λ2 given by 7 15(λ − 15)x2 ± 15(29λ + 165)(λ − 15) x3 . u(x) = (16.111) 150λ We now consider the following cases: 1. Using λ = 0 into (16.105) gives the trivial solution u(x) = 0. However, u(x) is undeﬁned by using λ = 0 into the last solution. Hence, λ = 0 is a singular point of equation (16.105). 2. There are two bifurcation points, namely λ = − 169 29 and λ = 15 for this equation. This shows that for λ < − 169 , and for λ > 15, then equation 29 (16.105) gives two real solutions, but has no real solutions for − 169 29 < λ < 15. Exercises 16.3.1

Use the direct computation method to solve the following Fredholm integro-diﬀerential equations: 1 1 x(u(t) − u3 (t)) dt, u(0) = 0 1. u (x) = λ 12 −1 1 1 λ 2. u (x) = x(u(t) − u2 (t)) dt, u(0) = 0 12 −1 1 1 x(1 − u2 (t)) dt, u(0) = 1 λ 3. u (x) = 24 0 π 1 4. u (x) = λ sin x sin t(1 + u2 (t)) dt, u(0) = 0 24 0 π 1 (x + x2 t)u2 (t) dt, u(0) = 0 5. u (x) = λ 12 0 π 6. u (x) = λ sin(x + t)(1 − u2 (t)) dt, u(0) = 0 7. u (x) = λ 8. u (x) = λ

9. u (x) = λ

−π π −π π −π

1

−1

cos(x + t)(1 − u2 (t)) dt, u(0) = 0 cos(x + t)(1 − u2 (t)) dt, u(0) = 0 x(u(t) + u2 (t)) dt, u(0) = 1, u (0) = 0

16.4 Systems of Nonlinear Fredholm Integro-Diﬀerential Equations 10. u (x) =

1 λ 2

1 λ 2 12. u (x) = λ

11. u (x) =

1

−1

1

−1 π

−π

535

x2 (1 + u2 (t)) dt, u(0) = 0, u (0) = 1 (1 − x2 t)(u(t) − u2 (t)) dt, u(0) = u (0) = 0

cos(x − t)(u(t) − u2 (t)) dt, u(0) = u (0) = 0

16.4 Systems of Nonlinear Fredholm Integro-Diﬀerential Equations In this section, systems of Fredholm integro-diﬀerential equations of the second kind given by b (i) ˜ 1 (x, t)F˜1 (v(t)) dt, u (x) = f1 (x) + K1 (x, t)F1 (u(t)) + K a (16.112) b ˜ 2 (x, t)F˜2 (v(t)) dt, K2 (x, t)F2 (u(t)) + K v(i) (x) = f2 (x) + a

will be studied. The unknown functions u(x) and v(x) occur inside the integral sign whereas the derivatives of u(x), v(x) appear mostly outside the ˜ i (x, t), and the function fi (x) are integral sign. The kernels Ki (x, t) and K given real-valued functions. The functions Fi and F˜i are nonlinear functions for u(x) and v(x) respectively. In Chapter 11, two analytical methods were used for solving systems of linear Fredholm integro-diﬀerential equations. These methods are the direct computation method and the variational iteration method. The aforementioned methods can eﬀectively handle the systems of nonlinear Fredholm integro-diﬀerential equations (16.112). The other methods presented in this text can also be used for handling such systems.

16.4.1 The Direct Computation Method The direct computation method will be applied to solve the systems of nonlinear Fredholm integro-diﬀerential equations of the second kind. The method approaches any Fredholm equation in a direct manner and gives the solution in an exact form and not in a series form. The method will be applied for the degenerate or separable kernels of the form n n ˜ k (t), ˜ K1 (x, t) = gk (x)hk (t), K1 (x, t) = g˜k (x)h K2 (x, t) =

k=1 n k=1

rk (x)sk (t),

˜ 2 (x, t) = K

k=1 n k=1

(16.113) r˜k (x)˜ sk (t).

536

16 Nonlinear Fredholm Integro-Diﬀerential Equations

The direct computation method can be applied as follows: 1. We ﬁrst substitute (16.113) into the system of Fredholm integrodiﬀerential equations (16.112) to obtain b b n n ˜ k (t)u(t)dt, u(i) (x) = f1 (x) + gk (x) hk (t)u(t)dt + g˜k (x) h k=1

v (i) (x) = f2 (x) +

n

a

k=1

b

rk (x)

sk (t)v(t) dt + a

k=1

n k=1

a b

r˜k (x)

s˜k (t)v(t) dt. a

(16.114) 2. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that Equation (16.114) becomes u(i) (x) = f1 (x) + α1 g1 (x) + · · · + αn gn (x) + β1 g˜1 (x) + · · · + βn g˜n (x), v (i) (x) = f2 (x) + γ1 r1 (x) + · · · + γn rn (x) + δ1 r˜1 (x) + · · · + δn r˜n (x), (16.115) where b b ˜ i (t) v(t)dt, 1 i n, hi (t) u(t)dt, 1 i n, βi = αi = h

a

b

si (t) u(t)dt, 1 i n,

γi = a

a b

s˜i (t) v(t)dt, 1 i n.

δi = a

(16.116) 3. Integrating both sides of (16.115) i times from 0 to x, and substituting the resulting equations for u(x) and v(x) into (16.116) gives a system of algebraic equations that can be solved to determine the constants αi , βi , γi , and δi . Using the obtained numerical values of these constants, the solutions u(x) and v(x) of the system (16.112) follow immediately. Example 16.17 Solve the system of nonlinear Fredholm integro-diﬀerential equations by using the direct computation method π 2

π3 u (x) = sin x + x cos x − u (t) + v 2 (t) dt, u(0) = 0 + 3 0 (16.117) π

2 π 2 u (t) − v (t) dt, v(0) = 0. v (x) = cos x − x sin x + + 2 0 Following the analysis presented above, this system can be rewritten as π3 , u (x) = sin x + x cos x + α + β − 3 (16.118) π , v (x) = cos x − x sin x + α − β + 2 where π π α= u2 (t) dt, β = v 2 (t) dt. (16.119) 0

0

16.4 Systems of Nonlinear Fredholm Integro-Diﬀerential Equations

537

Integrating both sides of (16.118) once from 0 to x gives π3 x, u(x) = x sin x + α + β − 3 (16.120) π x. v(x) = x cos x + α − β + 2 Substituting (16.120) into (16.119), and solving the resulting system gives 1 1 1 1 α = π3 − π, β = π3 + π. (16.121) 6 4 6 4 Substituting (16.121) into (16.120) leads to the exact solutions (u(x), v(x)) = (x sin x, x cos x).

(16.122)

Example 16.18 Solve the system of nonlinear Fredholm integro-diﬀerential equations by using the direct computation method ln 2 15 + (u2 (t) + v 2 (t))dt, u(0) = 1, u (x) = sinh x − 16 0 (16.123) v (x) = cosh x − ln 2 +

ln 2

0

(u2 (t) − v 2 (t))dt, v(0) = 0.

This system can be rewritten as

15 u (x) = sinh x + α + β − , 16 v (x) = cosh x + (α − β − ln 2),

where

α= 0

ln 2

u2 (t) dt,

β=

ln 2

v 2 (t) dt.

(16.124)

(16.125)

0

Integrating both sides of (16.124) once from 0 to x, and using the initial conditions we ﬁnd 15 x, u(x) = cosh x + α + β − 16 (16.126) v(x) = sinh x + (α − β − ln 2)x. To determine α, and β, we substitute (16.126) into (16.125) and solving the resulting system we obtain 15 1 15 1 + ln 2, β = − ln 2. (16.127) α= 32 2 32 2 Substituting (16.127) into (16.126) leads to the exact solutions (u(x), v(x)) = (cosh x, sinh x).

(16.128)

Example 16.19 Solve the system of nonlinear Fredholm integro-diﬀerential equations by using the direct computation method

538

16 Nonlinear Fredholm Integro-Diﬀerential Equations

u (x) = ex − 12 +

3x

v (x) = 3e

ln 2

(u2 (t) + v 2 (t))dt, u(0) = 1,

0

+9+ 0

ln 2

(16.129) 2

2

(u (t) − v (t))dt, v(0) = 1.

Proceeding as before we set u (x) = ex + (α + β − 12), v (x) = 3e3x + (α − β + 9), where ln 2

α=

u2 (t) dt,

0

ln 2

β=

v 2 (t) dt.

(16.130)

(16.131)

0

Integrating both sides of (16.130) once from 0 to x, and using the initial conditions we ﬁnd u(x) = ex + (α + β − 12)x, (16.132) v(x) = e3x + (α − β + 9)x. To determine α, and β, we substitute (16.132) into (16.131) and solving the resulting system we obtain 3 21 α= , β= . (16.133) 2 2 Substituting (16.133) into (16.132) leads to the exact solutions (u(x), v(x)) = (ex , e3x ).

(16.134)

Example 16.20 Solve the system of nonlinear Fredholm integro-diﬀerential equations by using the direct computation method π2 2 (v(t)w(t)) dt, u (x) = − sin x − + 3 0 u(0) = 0, u (0) = 1 π2 2 v (x) = − cos x − + (w(t)u(t)) dt, 3 0

(16.135)

v(0) = 1, v (0) = 0, π2 1 + (u(t)v(t)) dt, w (x) = −4 sin 2x − 2 0 w(0) = 0, w (0) = 2. This system can be rewritten as 2 , u (x) = − sin x + α − 3 2 , v (x) = − cos x + β − 3

(16.136)

16.4 Systems of Nonlinear Fredholm Integro-Diﬀerential Equations

w (x) = −4 sin 2x + γ − where

π 2

α=

1 2

539

,

v(t)w(t) dt,

0

β=

π 2

w(t)u(t) dt,

(16.137)

0

γ=

π 2

u(t)v(t) dt.

0

Integrating both sides of (16.136) twice from 0 to x, and using the initial conditions we obtain 1 1 α− x2 , u(x) = sin x + 2 3 1 1 β− x2 , v(x) = cos x + (16.138) 2 3 1 1 γ− x2 . w(x) = sin 2x + 2 4 Proceeding as before we obtain 2 1 2 (16.139) α= ,β= ,γ= . 3 3 2 This in turn gives the exact solutions (u(x), v(x), w(x)) = (sin x, cos x, sin 2x). (16.140)

Exercises 16.4.1 Use the direct computation method to solve the following systems of nonlinear Fredholm integro-diﬀerential equations π ⎧

⎪ 3 ⎪ 3 u2 (t) + v 2 (t) dt, u(0) = 0 ⎨ u (x) = cos x − x sin x − π + 0π 1.

2 π ⎪ ⎪ ⎩ v (x) = sin x + x cos x − + u (t) − v 2 (t) dt, v(0) = 0 2 0 ⎧ ln 2 2

15 1 ⎪ ⎪ u (x) = sinh 2x + (ln 2) + + u (t) − u(t)v(t) dt ⎪ ⎪ 2 32 ⎪ 0 ⎨ ln 2

2 3 15 2. ⎪ + v (t) − u(t)v(t) dt v (x) = sinh 2x − ln 2 − ⎪ ⎪ 2 32 ⎪ 0 ⎪ ⎩ u(0) = 1, v(0) = 2 π ⎧ π 4 ⎪ ⎪ (x) = − sin x − u2 (t) v 2 (t) dt, u(0) = 1 + u ⎨ 4 0 3. π ⎪ ⎪ ⎩ v (x) = sec x tan x − π + 2 u(t) v(t) dt, v(0) = 1 4 0

540

16 Nonlinear Fredholm Integro-Diﬀerential Equations

ln 2 ⎧

2 15 ⎪ ⎪ u (x) = sinh(2x) − (ln 2) − + u (t) + u(t)v(t) dt ⎪ ⎪ ⎪ 16 0 ⎨ ln 2

2 15 4. ⎪ + v (t) + u(t)v(t) dt v (x) = − sinh(2x) − 3 ln 2 + ⎪ ⎪ 16 ⎪ 0 ⎪ ⎩ u(0) = 1, v(0) = 1 ln 2 ⎧

x−2t 2 1 ⎪ ⎪ u (x) = (1 + x)ex − (ln 2)3 (ex − 1) + u (t) − u(t)v(t) dt, u(0) = 0 e ⎨ 3 0 5. ln 2 ⎪ x+2t 2

⎪ ⎩ v (x) = (1 − x)e−x − 1 (ln 2)3 (ex + 1) + e v (t) + u(t)v(t) dt, v(0) = 0 3 0 ⎧ π

π 2 ⎪ ⎪ u2 (t) + u(t)v(t) dt u (x) = − cos x − sin x − − 1 + ⎪ ⎪ 2 ⎪ 0 ⎨ π

π 6. 2 2 ⎪ v (t) + u(t)v(t) dt v (x) = − cos x + sin x − + 1 + ⎪ ⎪ 2 ⎪ 0 ⎪ ⎩ u(0) = 1, u (0) = 1, v(0) = 1, v (0) = −1 1 ⎧

2 1 ⎪ ⎪ u (x) = ex + − e2 + u (t) + v 2 (t) dt, u(0) = 1, u (0) = 2 ⎨ 3 0 7. 1

2 ⎪ ⎪ ⎩ v (x) = −ex − 4 + u (t) − v 2 (t) dt, v(0) = −1, v (0) = 0 0

ln 2 ⎧ 255 ⎪ x ⎪ (x) = e − v(t)w(t)dt, u(0) = 1, u (0) = 1 + u ⎪ ⎪ 8 ⎪ 0 ⎪ ⎪ ln 2 ⎨ 21 8. + v (x) = 9e3x − w(t)u(t)dt, v(0) = 1, v (0) = 3 ⎪ 2 ⎪ 0 ⎪ ln 2 ⎪ ⎪ ⎪ 15 ⎪ ⎩ w (x) = 25e5x − u(t)v(t)dt, w(0) = 1, w (0) = 5 + 4 0

16.4.2 The Variational Iteration Method The variational iteration method was used to handle the Fredholm integral equations, the Fredholm integro-diﬀerential equations and the nonlinear Fredholm integro-diﬀerential equations. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the so called Adomian polynomials that we need for nonlinear terms. The correction functionals for the system of nonlinear Fredholm integrodiﬀerential equations b (i) % 1 (x, t)F%1 (v(t)) dt, K1 (x, t)F1 (u(t)) + K u (x) = f1 (x) + a (16.141) b % 2 (x, t)F%2 (v(t)) dt. K2 (x, t)F2 (u(t)) + K v(i) (x) = f2 (x) + a

16.4 Systems of Nonlinear Fredholm Integro-Diﬀerential Equations

are given by

x

0 x

un+1 (x) = un (x) + vn+1 (x) = vn (x) +

b

Γ1 =

λ(t) u(i) n (t) − f1 (t) − Γ1 (t) dt, λ(t)

0

vn(i) (t)

− f2 (t) − Γ2 (t) dt.

Γ2 =

(16.142)

% 1 (t, r)F%1 (˜ K1 (t, r)F1 (˜ un (r)) + K vn (r)) dr,

a b

541

% 2 (t, r)F%2 (˜ un (r)) + K vn (r)) dr. K2 (t, r)F2 (˜

(16.143)

a

As stated before it is necessary to determine the Lagrange multiplier λ that can be identiﬁed optimally. An iteration formula, without restricted variation, should be used for the determination of the successive approximations un (x), vn (x), n 0 of the solutions u(x) and v(x). The zeroth approximations u0 (x) and v0 (x) can be selected by using the initial conditions. Consequently, the solutions are given by u(x) = lim un (x), v(x) = lim vn (x). n→∞

n→∞

(16.144)

The VIM will be illustrated by studying the following examples. Example 16.21 Use the variational iteration method to solve the system of nonlinear Fredholm integro-diﬀerential equations π u (x) = cos x − 5 sin x + cos(x − t)(u2 (t) + v 2 (t))dt, u(0) = 1, 0

v (x) = 3 cos x − sin x +

0

π

sin(x − t)(u2 (t) + v 2 (t))dt, v(0) = 1. (16.145)

The correction functionals for this system are given by x un+1 (x) = un (x) − (un (t) − cos t + 5 sin t − ρ1 ) dt, vn+1 (x) = vn (x) −

0

where ρ1 =

ρ2 =

0 x

π 0

(16.146) (vn (t)

cos(t − r)(u2n (r) + vn2 (r)) dr, (16.147)

π 0

− 3 cos t + sin t − ρ2 ) dt,

sin(t −

r)(u2n (r)

+

vn2 (r)) dr.

We can select the zeroth approximations u0 (x) = u(0) = 1 and v0 (x) = v(0) = 1. Using this selection, the correction functionals give the following successive approxim

Linear and Nonlinear Integral Equations Methods and Applications

Abdul-Majid Wazwaz

Linear and Nonlinear Integral Equations Methods and Applications

With 4 figures

~

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HIGHER EDUCATION PRESS

f1 Springer

THIS BOOK IS DEDICATED TO my wife, our son, and our three daughters for supporting me in all my endeavors

Preface

Many remarkable advances have been made in the ﬁeld of integral equations, but these remarkable developments have remained scattered in a variety of specialized journals. These new ideas and approaches have rarely been brought together in textbook form. If these ideas merely remain in scholarly journals and never get discussed in textbooks, then specialists and students will not be able to beneﬁt from the results of the valuable research achievements. The explosive growth in industry and technology requires constructive adjustments in mathematics textbooks. The valuable achievements in research work are not found in many of today’s textbooks, but they are worthy of addition and study. The technology is moving rapidly, which is pushing for valuable insights into some substantial applications and developed approaches. The mathematics taught in the classroom should come to resemble the mathematics used in varied applications of nonlinear science models and engineering applications. This book was written with these thoughts in mind. Linear and Nonlinear Integral Equations: Methods and Applications is designed to serve as a text and a reference. The book is designed to be accessible to advanced undergraduate and graduate students as well as a research monograph to researchers in applied mathematics, physical sciences, and engineering. This text diﬀers from other similar texts in a number of ways. First, it explains the classical methods in a comprehensible, non-abstract approach. Furthermore, it introduces and explains the modern developed mathematical methods in such a fashion that shows how the new methods can complement the traditional methods. These approaches further improve the understanding of the material. The book avoids approaching the subject through the compact and classical methods that make the material diﬃcult to be grasped, especially by students who do not have the background in these abstract concepts. The aim of this book is to oﬀer practical treatment of linear and nonlinear integral equations emphasizing the need to problem solving rather than theorem proving. The book was developed as a result of many years of experiences in teaching integral equations and conducting research work in this ﬁeld. The author

viii

Preface

has taken account of his teaching experience, research work as well as valuable suggestions received from students and scholars from a wide variety of audience. Numerous examples and exercises, ranging in level from easy to difﬁcult, but consistent with the material, are given in each section to give the reader the knowledge, practice and skill in linear and nonlinear integral equations. There is plenty of material in this text to be covered in two semesters for senior undergraduates and beginning graduates of mathematics, physical science, and engineering. The content of the book is divided into two distinct parts, and each part is self-contained and practical. Part I contains twelve chapters that handle the linear integral and nonlinear integro-diﬀerential equations by using the modern mathematical methods, and some of the powerful traditional methods. Since the book’s readership is a diverse and interdisciplinary audience of applied mathematics, physical science, and engineering, attempts are made so that part I presents both analytical and numerical approaches in a clear and systematic fashion to make this book accessible to those who work in these ﬁelds. Part II contains the remaining six chapters devoted to thoroughly examining the nonlinear integral equations and its applications. The potential theory contributed more than any ﬁeld to give rise to nonlinear integral equations. Mathematical physics models, such as diﬀraction problems, scattering in quantum mechanics, conformal mapping, and water waves also contributed to the creation of nonlinear integral equations. Because it is not always possible to ﬁnd exact solutions to problems of physical science that are posed, much work is devoted to obtaining qualitative approximations that highlight the structure of the solution. Chapter 1 provides the basic deﬁnitions and introductory concepts. The Taylor series, Leibnitz rule, and Laplace transform method are presented and reviewed. This discussion will provide the reader with a strong basis to understand the thoroughly-examined material in the following chapters. In Chapter 2, the classiﬁcations of integral and integro-diﬀerential equations are presented and illustrated. In addition, the linearity and the homogeneity concepts of integral equations are clearly addressed. The conversion process of IVP and BVP to Volterra integral equation and Fredholm integral equation respectively are described. Chapters 3 and 5 deal with the linear Volterra integral equations and the linear Volterra integro-diﬀerential equations, of the ﬁrst and the second kind, respectively. Each kind is approached by a variety of methods that are described in details. Chapters 3 and 5 provide the reader with a comprehensive discussion of both types of equations. The two chapters emphasize the power of the proposed methods in handling these equations. Chapters 4 and 6 are entirely devoted to Fredholm integral equations and Fredholm integro-diﬀerential equations, of the ﬁrst and the second kind, respectively. The ill-posed Fredholm integral equation of the ﬁrst kind is handled by the powerful method of regularization combined with other methods. The two kinds of equations are approached

Preface

ix

by many appropriate algorithms that are illustrated in details. A comprehensive study is introduced where a variety of reliable methods is applied independently and supported by many illustrative examples. Chapter 7 is devoted to studying the Abel’s integral equations, generalized Abel’s integral equations, and the weakly singular integral equations. The chapter also stresses the signiﬁcant features of these types of singular equations with full explanations and many illustrative examples and exercises. Chapters 8 and 9 introduce a valuable study on Volterra-Fredholm integral equations and Volterra-Fredholm integro-diﬀerential equations respectively in one and two variables. The mixed Volterra-Fredholm integral and the mixed VolterraFredholm integro-diﬀerential equations in two variables are also examined with illustrative examples. The proposed methods introduce a powerful tool for handling these two types of equations. Examples are provided with a substantial amount of explanation. The reader will ﬁnd a wealth of well-known models with one and two variables. A detailed and clear explanation of every application is introduced and supported by fully explained examples and exercises of every type. Chapters 10, 11, and 12 are concerned with the systems of Volterra integral and integro-diﬀerential equations, systems of Fredholm integral and integro-diﬀerential equations, and systems of singular integral equations and systems of weakly singular integral equations respectively. Systems of integral equations that are important, are handled by using very constructive methods. A discussion of the basic theory and illustrations of the solutions to the systems are presented to introduce the material in a clear and useful fashion. Singular systems in one, two, and three variables are thoroughly investigated. The systems are supported by a variety of useful methods that are well explained and illustrated. Part II is titled “Nonlinear Integral Equations”. Part II of this book gives a self-contained, practical and realistic approach to nonlinear integral equations, where scientists and engineers are paying great attention to the eﬀects caused by the nonlinearity of dynamical equations in nonlinear science. The potential theory contributed more than any ﬁeld to give rise to nonlinear integral equations. Mathematical physics models, such as diﬀraction problems, scattering in quantum mechanics, conformal mapping, and water waves also contributed to the creation of nonlinear integral equations. The nonlinearity of these models may give more than one solution and this is the nature of nonlinear problems. Moreover, ill-posed Fredholm integral equations of the ﬁrst kind may also give more than one solution even if it is linear. Chapter 13 presents discussions about nonlinear Volterra integral equations and systems of Volterra integral equations, of the ﬁrst and the second kind. More emphasis on the existence of solutions is proved and emphasized. A variety of methods are employed, introduced and explained in a clear and useful manner. Chapter 14 is devoted to giving a comprehensive study on nonlinear Volterra integro-diﬀerential equations and systems of nonlinear Volterra integro-diﬀerential equations, of the ﬁrst and the second kind.

x

Preface

A variety of methods are introduced, and numerous practical examples are explained in a practical way. Chapter 15 investigates thoroughly the existence theorem, bifurcation points and singular points that may arise from nonlinear Fredholm integral equations. The study presents a variety of powerful methods to handle nonlinear Fredholm integral equations of the ﬁrst and the second kind. Systems of these equations are examined with illustrated examples. Chapter 16 is entirely devoted to studying a family of nonlinear Fredholm integro-diﬀerential equations of the second kind and the systems of these equations. The approach we followed is identical to our approach in the previous chapters to make the discussion accessible for interdisciplinary audience. Chapter 17 provides the reader with a comprehensive discussion of the nonlinear singular integral equations, nonlinear weakly singular integral equations, and systems of these equations. Most of these equations are characterized by the singularity behavior where the proposed methods should overcome the diﬃculty of this singular behavior. The power of the employed methods is conﬁrmed here by determining solutions that may not be unique. Chapter 18 presents a comprehensive study on ﬁve scientiﬁc applications that we selected because of its wide applicability for several other models. Because it is not always possible to ﬁnd exact solutions to models of physical sciences, much work is devoted to obtaining qualitative approximations that highlight the structure of the solution. The powerful Pad´e approximants are used to give insight into the structure of the solution. This chapter closes Part II of this text. The book concludes with seven useful appendices. Moreover, the book introduces the traditional methods in the same amount of concern to provide the reader with the knowledge needed to make a comparison. I deeply acknowledge Professor Albert Luo for many helpful discussions, encouragement, and useful remarks. I am also indebted to Ms. Liping Wang, the Publishing Editor of the Higher Education Press for her eﬀective cooperation and important suggestions. The staﬀ of HEP deserve my thanks for their support to this project. I owe them all my deepest thanks. I also deeply acknowledge Professor Louis Pennisi who made very valuable suggestions that helped a great deal in directing this book towards its main goal. I am deeply indebted to my wife, my son and my daughters who provided me with their continued encouragement, patience and support during the long days of preparing this book. The author would highly appreciate any notes concerning any constructive suggestions.

Abdul-Majid Wazwaz Saint Xavier University Chicago, IL 60655 April 20, 2011

Contents

Part I Linear Integral Equations 1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Ordinary Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 First Order Linear Diﬀerential Equations . . . . . . . . . . . . 1.2.2 Second Order Linear Diﬀerential Equations . . . . . . . . . . 1.2.3 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 1.3 Leibnitz Rule for Diﬀerentiation of Integrals . . . . . . . . . . . . . . . 1.4 Reducing Multiple Integrals to Single Integrals . . . . . . . . . . . . . 1.5 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Properties of Laplace Transforms . . . . . . . . . . . . . . . . . . . 1.6 Inﬁnite Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 4 7 7 9 13 17 19 22 23 28 30

2

Introductory Concepts of Integral Equations . . . . . . . . . . . . . . 2.1 Classiﬁcation of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Fredholm Integral Equations . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Volterra-Fredholm Integral Equations . . . . . . . . . . . . . . . 2.1.4 Singular Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Classiﬁcation of Integro-Diﬀerential Equations . . . . . . . . . . . . . 2.2.1 Fredholm Integro-Diﬀerential Equations . . . . . . . . . . . . . 2.2.2 Volterra Integro-Diﬀerential Equations . . . . . . . . . . . . . . 2.2.3 Volterra-Fredholm Integro-Diﬀerential Equations . . . . . 2.3 Linearity and Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Linearity Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Homogeneity Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Origins of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Converting IVP to Volterra Integral Equation . . . . . . . . . . . . . .

33 34 34 35 35 36 37 38 38 39 40 40 41 42 42

xii

Contents

2.5.1 Converting Volterra Integral Equation to IVP . . . . . . . . 2.6 Converting BVP to Fredholm Integral Equation . . . . . . . . . . . . 2.6.1 Converting Fredholm Integral Equation to BVP . . . . . . 2.7 Solution of an Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 49 54 59 63

Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Volterra Integral Equations of the Second Kind . . . . . . . . . . . . 3.2.1 The Adomian Decomposition Method . . . . . . . . . . . . . . . 3.2.2 The Modiﬁed Decomposition Method . . . . . . . . . . . . . . . 3.2.3 The Noise Terms Phenomenon . . . . . . . . . . . . . . . . . . . . . 3.2.4 The Variational Iteration Method . . . . . . . . . . . . . . . . . . 3.2.5 The Successive Approximations Method . . . . . . . . . . . . . 3.2.6 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . 3.2.7 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 3.3 Volterra Integral Equations of the First Kind . . . . . . . . . . . . . . 3.3.1 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . 3.3.3 Conversion to a Volterra Equation of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66 66 73 78 82 95 99 103 108 108 111

4

Fredholm Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fredholm Integral Equations of the Second Kind . . . . . . . . . . . 4.2.1 The Adomian Decomposition Method . . . . . . . . . . . . . . . 4.2.2 The Modiﬁed Decomposition Method . . . . . . . . . . . . . . . 4.2.3 The Noise Terms Phenomenon . . . . . . . . . . . . . . . . . . . . . 4.2.4 The Variational Iteration Method . . . . . . . . . . . . . . . . . . 4.2.5 The Direct Computation Method . . . . . . . . . . . . . . . . . . . 4.2.6 The Successive Approximations Method . . . . . . . . . . . . . 4.2.7 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 4.3 Homogeneous Fredholm Integral Equation . . . . . . . . . . . . . . . . . 4.3.1 The Direct Computation Method . . . . . . . . . . . . . . . . . . . 4.4 Fredholm Integral Equations of the First Kind . . . . . . . . . . . . . 4.4.1 The Method of Regularization . . . . . . . . . . . . . . . . . . . . . 4.4.2 The Homotopy Perturbation Method . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 121 121 128 133 136 141 146 151 154 155 159 161 166 173

5

Volterra Integro-Diﬀerential Equations . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Volterra Integro-Diﬀerential Equations of the Second Kind . . . 5.2.1 The Adomian Decomposition Method . . . . . . . . . . . . . . . 5.2.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . .

175 175 176 176 181

3

114 118

Contents

5.2.3 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . 5.2.4 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Converting Volterra Integro-Diﬀerential Equations to Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Converting Volterra Integro-Diﬀerential Equation to Volterra Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Volterra Integro-Diﬀerential Equations of the First Kind . . . . . 5.3.1 Laplace Transform Method . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

7

8

Fredholm Integro-Diﬀerential Equations . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 The Direct Computation Method . . . . . . . . . . . . . . . . . . . 6.2.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . . 6.2.3 The Adomian Decomposition Method . . . . . . . . . . . . . . . 6.2.4 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abel’s Integral Equation and Singular Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Abel’s Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . 7.3 The Generalized Abel’s Integral Equation . . . . . . . . . . . . . . . . . 7.3.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . 7.3.2 The Main Generalized Abel Equation . . . . . . . . . . . . . . . 7.4 The Weakly Singular Volterra Equations . . . . . . . . . . . . . . . . . . 7.4.1 The Adomian Decomposition Method . . . . . . . . . . . . . . . 7.4.2 The Successive Approximations Method . . . . . . . . . . . . . 7.4.3 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volterra-Fredholm Integral Equations . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Volterra-Fredholm Integral Equations . . . . . . . . . . . . . . . . . 8.2.1 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 The Adomian Decomposition Method . . . . . . . . . . . . . . . 8.3 The Mixed Volterra-Fredholm Integral Equations . . . . . . . . . . . 8.3.1 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 The Adomian Decomposition Method . . . . . . . . . . . . . . . 8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

186 190 195 199 203 204 206 211 213 213 214 214 218 223 230 234 237 237 238 239 242 243 245 247 248 253 257 260 261 261 262 262 266 269 270 273 277

xiv

Contents

8.4.1 The Modiﬁed Decomposition Method . . . . . . . . . . . . . . . 278 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Volterra-Fredholm Integro-Diﬀerential Equations . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Volterra-Fredholm Integro-Diﬀerential Equation . . . . . . . . 9.2.1 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . . 9.3 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 The Direct Computation Method . . . . . . . . . . . . . . . . . . . 9.3.2 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . 9.4 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 The Modiﬁed Decomposition Method . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

285 285 285 285 289

10 Systems of Volterra Integral Equations . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Systems of Volterra Integral Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 The Adomian Decomposition Method . . . . . . . . . . . . . 10.2.2 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . 10.3 Systems of Volterra Integral Equations of the First Kind . . . 10.3.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . 10.3.2 Conversion to a Volterra System of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Systems of Volterra Integro-Diﬀerential Equations . . . . . . . . . 10.4.1 The Variational Iteration Method . . . . . . . . . . . . . . . . . 10.4.2 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311 311

11 Systems of Fredholm Integral Equations . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Systems of Fredholm Integral Equations . . . . . . . . . . . . . . . . . . 11.2.1 The Adomian Decomposition Method . . . . . . . . . . . . . 11.2.2 The Direct Computation Method . . . . . . . . . . . . . . . . . 11.3 Systems of Fredholm Integro-Diﬀerential Equations . . . . . . . . 11.3.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 11.3.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341 341 342 342 347 352 353 358 364

9

296 296 300 303 304 309

312 312 318 323 323 327 328 329 335 339

Contents

12 Systems of Singular Integral Equations . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Systems of Generalized Abel Integral Equations . . . . . . . . . . . 12.2.1 Systems of Generalized Abel Integral Equations in Two Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Systems of Generalized Abel Integral Equations in Three Unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Systems of the Weakly Singular Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . 12.3.2 The Adomian Decomposition Method . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

365 365 366 366 370 374 374 378 383

Part II Nonlinear Integral Equations 13 Nonlinear Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Existence of the Solution for Nonlinear Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Nonlinear Volterra Integral Equations of the Second Kind . . 13.3.1 The Successive Approximations Method . . . . . . . . . . . 13.3.2 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . 13.3.3 The Adomian Decomposition Method . . . . . . . . . . . . . 13.4 Nonlinear Volterra Integral Equations of the First Kind . . . . 13.4.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . 13.4.2 Conversion to a Volterra Equation of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Systems of Nonlinear Volterra Integral Equations . . . . . . . . . . 13.5.1 Systems of Nonlinear Volterra Integral Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Systems of Nonlinear Volterra Integral Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Nonlinear Volterra Integro-Diﬀerential Equations . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 The Combined Laplace Transform-Adomian Decomposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . 14.2.3 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . 14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

387 387 388 388 389 393 397 404 405 408 411 412 417 423 425 425 426 426 432 436 440

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Contents

14.3.1 The Combined Laplace Transform-Adomian Decomposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Conversion to Nonlinear Volterra Equation of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 The Variational Iteration Method . . . . . . . . . . . . . . . . . 14.4.2 The Combined Laplace Transform-Adomian Decomposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

440 446 450 451 456 465

15 Nonlinear Fredholm Integral Equations . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Existence of the Solution for Nonlinear Fredholm Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Bifurcation Points and Singular Points . . . . . . . . . . . . . 15.3 Nonlinear Fredholm Integral Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 15.3.2 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . 15.3.3 The Adomian Decomposition Method . . . . . . . . . . . . . 15.3.4 The Successive Approximations Method . . . . . . . . . . . 15.4 Homogeneous Nonlinear Fredholm Integral Equations . . . . . . 15.4.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 15.5 Nonlinear Fredholm Integral Equations of the First Kind . . . 15.5.1 The Method of Regularization . . . . . . . . . . . . . . . . . . . . 15.5.2 The Homotopy Perturbation Method . . . . . . . . . . . . . . 15.6 Systems of Nonlinear Fredholm Integral Equations . . . . . . . . . 15.6.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 15.6.2 The Modiﬁed Adomian Decomposition Method . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

467 467

16 Nonlinear Fredholm Integro-Diﬀerential Equations . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Nonlinear Fredholm Integro-Diﬀerential Equations . . . . . . . . . 16.2.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 16.2.2 The Variational Iteration Method . . . . . . . . . . . . . . . . . 16.2.3 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . 16.3 Homogeneous Nonlinear Fredholm Integro-Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 16.4 Systems of Nonlinear Fredholm Integro-Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 The Direct Computation Method . . . . . . . . . . . . . . . . . 16.4.2 The Variational Iteration Method . . . . . . . . . . . . . . . . .

517 517 518 518 522 526

468 469 469 470 476 480 485 490 490 494 495 500 505 506 510 515

530 530 535 535 540

Contents

xvii

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 17 Nonlinear Singular Integral Equations . . . . . . . . . . . . . . . . . . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Nonlinear Abel’s Integral Equation . . . . . . . . . . . . . . . . . . . . . . 17.2.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . 17.3 The Generalized Nonlinear Abel Equation . . . . . . . . . . . . . . . . 17.3.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . 17.3.2 The Main Generalized Nonlinear Abel Equation . . . . 17.4 The Nonlinear Weakly-Singular Volterra Equations . . . . . . . . 17.4.1 The Adomian Decomposition Method . . . . . . . . . . . . . 17.5 Systems of Nonlinear Weakly-Singular Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5.1 The Modiﬁed Adomian Decomposition Method . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

547 547 548 549 552 553 556 559 559

18 Applications of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Volterra’s Population Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 The Variational Iteration Method . . . . . . . . . . . . . . . . . 18.2.2 The Series Solution Method . . . . . . . . . . . . . . . . . . . . . . 18.2.3 The Pad´e Approximants . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Integral Equations with Logarithmic Kernels . . . . . . . . . . . . . . 18.3.1 Second Kind Fredholm Integral Equation with a Logarithmic Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.2 First Kind Fredholm Integral Equation with a Logarithmic Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.3 Another First Kind Fredholm Integral Equation with a Logarithmic Kernel . . . . . . . . . . . . . . . . . . . . . . . 18.4 The Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5 The Thomas-Fermi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6 Heat Transfer and Heat Radiation . . . . . . . . . . . . . . . . . . . . . . . 18.6.1 Heat Transfer: Lighthill Singular Integral Equation . . 18.6.2 Heat Radiation in a Semi-Inﬁnite Solid . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

569 569 570 571 572 573 574

583 584 587 590 590 592 594

Appendix A Table of Indeﬁnite Integrals . . . . . . . . . . . . . . . . . . . . A.1 Basic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Trigonometric Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Inverse Trigonometric Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Exponential and Logarithmic Forms . . . . . . . . . . . . . . . . . . . . . . A.5 Hyperbolic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Other Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

597 597 597 598 598 599 599

562 563 567

577 580

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Contents

Appendix B B.1

Integrals Involving Irrational Algebraic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 n Integrals Involving √tx−t , n is an integer, n 0 . . . . . . . . . . . . 600

B.2

Integrals Involving

n

2 √t , x−t

n is an odd integer, n 1 . . . . . . . . 600

Appendix C Series Representations . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Exponential Functions Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . C.4 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5 Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.6 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

601 601 601 602 602 602 602

Appendix D The Error and the Complementary Error Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 D.1 The Error Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 D.2 The Complementary Error Function . . . . . . . . . . . . . . . . . . . . . . 603 Appendix E

Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

Appendix F Inﬁnite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 F.1 Numerical Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 F.2 Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Appendix G The Fresnel Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 G.1 The Fresnel Cosine Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 G.2 The Fresnel Sine Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

Part I

Linear Integral Equations

Chapter 1

Preliminaries

An integral equation is an equation in which the unknown function u(x) appears under an integral sign [1–7]. A standard integral equation in u(x) is of the form: h(x)

u(x) = f (x) + λ

K(x, t)u(t)dt,

(1.1)

g(x)

where g(x) and h(x) are the limits of integration, λ is a constant parameter, and K(x, t) is a function of two variables x and t called the kernel or the nucleus of the integral equation. The function u(x) that will be determined appears under the integral sign, and it appears inside the integral sign and outside the integral sign as well. The functions f (x) and K(x, t) are given in advance. It is to be noted that the limits of integration g(x) and h(x) may be both variables, constants, or mixed. An integro-diﬀerential equation is an equation in which the unknown function u(x) appears under an integral sign and contains an ordinary derivative u(n) (x) as well. A standard integro-diﬀerential equation is of the form: h(x) K(x, t)u(t)dt, (1.2) u(n) (x) = f (x) + λ g(x)

where g(x), h(x), f (x), λ and the kernel K(x, t) are as prescribed before. Integral equations and integro-diﬀerential equations will be classiﬁed into distinct types according to the limits of integration and the kernel K(x, t). All types of integral equations and integro-diﬀerential equations will be classiﬁed and investigated in the forthcoming chapters. In this chapter, we will review the most important concepts needed to study integral equations. The traditional methods, such as Taylor series method and the Laplace transform method, will be used in this text. Moreover, the recently developed methods, that will be used thoroughly in this text, will determine the solution in a power series that will converge to an exact solution if such a solution exists. However, if exact solution does not exist, we use as many terms of the obtained series for numerical purposes to approximate the solution. The more terms we determine the higher numerical A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

4

1 Preliminaries

accuracy we can achieve. Furthermore, we will review the basic concepts for solving ordinary diﬀerential equations. Other mathematical concepts, such as Leibnitz rule will be presented.

1.1 Taylor Series Let f (x) be a function with derivatives of all orders in an interval [x0 , x1 ] that contains an interior point a. The Taylor series of f (x) generated at x = a is ∞ f (n) (a) (x − a)n , (1.3) f (x) = n! n=0 or equivalently f (a) f (a) f (a) (x − a) + (x − a)2 + (x − a)3 + · · · 1! 2! 3! (1.4) f (n) (a) (x − a)n + · · · . + n! The Taylor series generated by f (x) at a = 0 is called the Maclaurin series and given by ∞ f (n) (0) n x , f (x) = (1.5) n! n=0 f (x) = f (a) +

that is equivalent to f (0) 2 f (0) 3 f (n) (0) n f (0) x+ x + x +···+ x + · · · . (1.6) f (x) = f (0) + 1! 2! 3! n! In what follows, we will discuss a few examples for the determination of the Taylor series at x = 0. Example 1.1 Find the Taylor series generated by f (x) = ex at x = 0. We list the exponential function and its derivatives as follows: f (n) (x) f (x) = ex f (x) = ex f (x) = ex f (x) = ex f (n) (0) f (0) = 1, f (0) = 1, f (0) = 1, .. . and so on. This gives the Taylor series for ex by x3 x4 x2 + + + ··· ex = 1 + x + 2! 3! 4! and in a compact form by ∞ xn ex = . n! n=0 Similarly, we can easily show that

f (0) = 1,

(1.7)

(1.8)

1.1 Taylor Series

5

x3 x4 x2 − + + ··· 2! 3! 4!

(1.9)

(ax)3 (ax)4 (ax)2 + + + ··· 2! 3! 4!

(1.10)

e−x = 1 − x + and eax = 1 + ax + Example 1.2

Find the Taylor series generated by f (x) = cos x at x = 0. Following the discussions presented before we ﬁnd f (n) (x) f (x) = cos x,f (x) = − sin x,f (x) = − cos x,f (x) = sin x,f (iv) (x) = cos x f (n) (0) f (0) = 1, .. .

f (0) = 0,

f (0) = −1,

f (0) = 0,

f (iv) (0) = 1,

and so on. This gives the Taylor series for cos x by x4 x2 + + ··· cos x = 1 − 2! 4! and in a compact form by ∞ (−1)n 2n cos x = x . (2n)! n=0

(1.11)

(1.12)

In a similar way we can derive the following cos(ax) = 1 −

∞ (−1)n (ax)4 (ax)2 + + ··· = (ax)2n . 2! 4! (2n)! n=0

For f (x) = sin x and f (x) = sin(ax), we can show that ∞ (−1)n 2n+1 x3 x5 sin x = x − + + ··· = x , 3! 5! (2n + 1)! n=0 sin(ax) = (ax) −

∞ (ax)5 (−1)n (ax)3 + + ··· = (ax)2n+1 . 3! 5! (2n + 1)! n=0

(1.13)

(1.14)

In Appendix C, the Taylor series for many well known functions generated at x = 0 are given. As stated before, the newly developed methods for solving integral equations determine the solution in a series form. Unlike calculus where we determine the Taylor series for functions and the radius of convergence for each series, it is required here that we determine the exact solution of the integral equation if we determine its series solution. In what follows, we will discuss some examples that will show how exact solution is obtained if the series solution is given. Recall that the Taylor series exists for analytic functions only. Example 1.3 Find the closed form function for the following series:

6

1 Preliminaries

4 2 f (x) = 1 + 2x + 2x2 + x3 + x4 + · · · 3 3 It is obvious that this series can be rewritten in the form: (2x)3 (2x)4 (2x)2 + + + ··· f (x) = 1 + 2x + 2! 3! 4! that will converge to the exact form: f (x) = e2x .

(1.15)

(1.16) (1.17)

Example 1.4 Find the closed form function for the following series: 9 27 81 f (x) = 1 + x2 + x4 + x6 + · · · 2 8 80

(1.18)

2

, therefore the series can Notice that the second term can be written as (3x) 2! be rewritten as (3x)2 (3x)4 (3x)6 f (x) = 1 + + + + ··· (1.19) 2! 4! 6! that will converge to f (x) = cosh(3x). (1.20) Example 1.5 Find the closed form function for the following series: 1 2 f (x) = x + x3 + x5 + · · · 3 15 This series will converge to f (x) = tan x.

(1.21) (1.22)

Example 1.6 Find the closed form function for the following series: 1 1 1 1 (1.23) f (x) = 1 + x − x2 − x3 + x4 + x5 + · · · 2! 3! 4! 5! The signs of the terms are positive for the ﬁrst two terms then negative for the next two terms and so on. The series should be grouped as 1 1 1 1 (1.24) f (x) = (1 − x2 + x4 + · · · ) + (x − x3 + x5 + · · · ), 2! 4! 3! 5! that will converge to f (x) = cos x + sin x. (1.25)

Exercises 1.1 Find the closed form function for the following Taylor series: 1. f (x) = 2x + 2x2 +

4 3 2 4 9 9 27 4 x + x + · · · 2. f (x) = 1 − 3x + x2 − x3 + x + ··· 3 3 2 2 8

1.2 Ordinary Diﬀerential Equations

7

1 2 1 1 x + x3 + x4 + · · · 2! 3! 4! 1 1 1 1 1 4. f (x) = 1 − x + x2 + x3 + x4 − x5 − x6 + · · · 2! 3! 4! 5! 6! 9 81 5 243 7 5. f (x) = 3x − x3 + x − x + ··· 2 40 560 4 4 5 8 7 6. f (x) = 2x + x3 + x + x + ··· 3 15 315 2 4 6 9 27 4 81 6 8. f (x) = x2 + 7. f (x) = 1 + 2x2 + x4 + x + ··· x + x + ··· 3 45 2 8 80 2 4 6 9. f (x) = 2 − 2 x2 + x4 − x + ··· 3 45 1 5 1 1 x − x7 + · · · 10. f (x) = 1 + x − x3 + 6 120 5040 1 2 5 17 7 x + x + ··· 11. f (x) = x + x3 + 3 15 315 2 5 17 7 1 x − x + ··· 12. f (x) = x − x3 + 3 15 315 1 1 1 1 13. f (x) = 2 + 2x + x2 − x3 + x4 + x5 + · · · . 2! 3! 4! 5! 1 1 1 14. 2 + x − x2 + x4 − x6 + · · · 2! 4! 6! 3. f (x) = x +

1.2 Ordinary Diﬀerential Equations In this section we will review some of the linear ordinary diﬀerential equations that we will use for solving integral equations. For proofs, existence and uniqueness of solutions, and other details, the reader is advised to use ordinary diﬀerential equations texts.

1.2.1 First Order Linear Diﬀerential Equations The standard form of ﬁrst order linear ordinary diﬀerential equation is u + p(x)u = q(x), (1.26) where p(x) and q(x) are given continuous functions on x0 < x < x1 . We ﬁrst determine an integrating factor μ(x) by using the formula: μ(x) = e

x

p(t)dt

.

(1.27)

Recall that an integrating factor μ(x) is a function of x that is used to facilitate the solving of a diﬀerential equation. The solution of (1.26) is obtained by using the formula: x 1 u(x) = μ(t)q(t)dt + c , (1.28) μ(x)

8

1 Preliminaries

where c is an arbitrary constant that can be determined by using a given initial condition. Example 1.7 Solve the following ﬁrst order ODE: u − 3u = 3x2 e3x ,

u(0) = 1.

(1.29)

2 3x

Notice that p(x) = −3 and q(x) = 3x e . The integrating factor μ(x) is obtained by x μ(x) = e 0 −3dt = e−3x . (1.30) Consequently, u(x) is obtained by using x 1 3x u(x) = μ(t)q(t)dt + c = e μ(x)

= e3x x3 + c = e3x (x3 + 1), obtained upon using the given initial condition.

x

2

3t dt + c

(1.31)

Example 1.8 Solve the following ﬁrst order ODE: cos x , u(π) = 0, x > 0. (1.32) xu + 3u = x We ﬁrst divide the equation by x to convert it to the standard form (1.26). x . The integrating factor μ(x) is As a result, p(x) = x3 and q(x) = cos x2 μ(x) = e

x

3 t dt

= e3 ln x = x3 .

Consequently, u(x) is x

1 1 x u(x) = μ(t)q(t)dt + c = 3 t cos tdt + c μ(x) x 1 1 = 3 (cos x + x sin x + c) = 3 (cos x + x sin x + 1), x x obtained upon using the given initial condition.

(1.33)

(1.34)

Exercises 1.2.1 Find the general solution for each of the following ﬁrst order ODEs: 1. u + u = e−x , x > 0

2. xu − 4u = x5 ex , x > 0

3. (x2 + 9)u + 2xu = 0, x > 0

4. xu − 4u = 2x6 + x5 , x > 0

5. xu + u = 2x, x > 0

6. xu − u = x2 sin x, x > 0

Find the particular solution for each of the following initial value problems: 7. u − u = 2xex , u(0) = 0

π π = e2 9. (tan x)u + (sec2 x)u = 2e2x , u 4

8. xu + u = 2x, u(1) = 1 10. u − 3u = 4x3 e3x , u(0) = 1

1.2 Ordinary Diﬀerential Equations 11. (1 + x3 )u + 3x2 u = 1, u(0) = 0

9 12. u + (tan x)u = cos x, u(0) = 1

1.2.2 Second Order Linear Diﬀerential Equations As stated before, we will review some second order linear ordinary diﬀerential equations. The focus will be on second order equations, homogeneous and inhomogeneous as well. Homogeneous Equations with Constant Coeﬃcients The standard form of the second order homogeneous ordinary diﬀerential equations with constant coeﬃcients is au + bu + cu = 0, a = 0, (1.35) where a, b, and c are constants. The solution of this equation is assumed to be of the form: u(x) = erx . (1.36) Substituting this assumption into Eq. (1.35) gives the equation: erx (ar2 + br + c) = 0. (1.37) rx Since e is not zero, then we have the characteristic or the auxiliary equation: (1.38) ar2 + br + c = 0. Solving this quadratic equation leads to one of the following three cases: (i) If the roots r1 and r2 are real and r1 = r2 , then the general solution of the homogeneous equation is (1.39) u(x) = Aer1 x + Ber2 x , where A and B are constants. (ii) If the roots r1 and r2 are real and r1 = r2 = r, then the general solution of the homogeneous equation is u(x) = Aerx + Bxerx , (1.40) where A and B are constants. (iii) If the roots r1 and r2 are complex and r1 = λ + iμ, r2 = λ − iμ, then the general solution of the homogeneous equation is given by u(x) = eλx (A cos(μx) + B sin(μx)) , where A and B are constants.

(1.41)

Inhomogeneous Equations with Constant Coeﬃcients The standard form of the second order inhomogeneous ordinary diﬀerential equations with constant coeﬃcients is

10

1 Preliminaries

au + bu + cu = g(x), a = 0,

(1.42)

where a, b, and c are constants. The general solution consists of two parts, namely, complementary solution uc , and a particular solution up where the general solution is of the form: u(x) = uc (x) + up (x), where uc is the solution of the related homogeneous equation:

(1.43)

(1.44) au + bu + cu = 0, a = 0, and this is obtained as presented before. A particular solution up arises from the inhomogeneous part g(x). It is called a particular solution because it justiﬁes the inhomogeneous equation (1.42), but it is not the particular solution of the equation that is obtained from (1.43) upon using the given initial equations as will be discussed later. To obtain up (x), we use the method of undetermined coeﬃcients. To apply this method, we consider the following three types of g(x): (i) If g(x) is a polynomial given by g(x) = a0 xn + a1 xn−1 + · · · + an , then up should be assumed as up = xr (b0 xn + b1 xn−1 + · · · + bn ), r = 0, 1, 2, . . . (ii) If g(x) is an exponential function of the form: g(x) = a0 eαx ,

(1.45) (1.46) (1.47)

then up should be assumed as up = b0 xr eαx , r = 0, 1, 2, . . . (iii) If g(x) is a trigonometric function of the form:

(1.48)

g(x) = a0 sin(αx) + b0 cos(βx), then up should be assumed as

(1.49)

(1.50) up = xr (A0 sin(αx) + B0 cos(βx)), r = 0, 1, 2, . . . For other forms of g(x) such as tan x and sec x, we usually use the variation of parameters method that will not be reviewed in this text. Notice that r is the smallest nonnegative integer that will guarantee no term in up (x) is a solution of the corresponding homogeneous equation. The values of r are 0, 1 and 2. Example 1.9 Solve the following second order ODE: u − u = 0.

(1.51)

The auxiliary equation is given by r2 − 1 = 0,

(1.52)

and this gives r = ±1.

(1.53)

1.2 Ordinary Diﬀerential Equations

11

Accordingly, the general solution is given by u(x) = Aex + Be−x . It is interesting to point out that the normal form ODE:

(1.54)

u + u = 0,

(1.55)

r2 + 1 = 0,

(1.56)

r = ±i.

(1.57)

leads to the auxiliary equation: and this gives The general solution is given by u(x) = A cos x + B sin x.

(1.58)

Example 1.10 Solve the following second order ODE: u − 7u + 6u = 0.

(1.59)

The auxiliary equation is given by r2 − 7r + 6 = 0,

(1.60)

with roots given by r = 1, 6. The general solution is given by u(x) = Aex + Be6x .

(1.61) (1.62)

Example 1.11 Solve the following second order ODE: (1.63) u − 5u + 6u = 6x + 7. We ﬁrst ﬁnd uc . The auxiliary equation for the related homogeneous equation is given by r2 − 5r + 6 = 0, (1.64) with roots given by r = 2, 3. The general solution is given by u(x) = αe2x + βe3x .

(1.65) (1.66)

Noting that g(x) = 6x + 7, then a particular solution is assumed to be of the form (1.67) up = Ax + B. Since this is a particular solution, then substituting up into the inhomogeneous equation leads to 6Ax + (6B − 5A) = 6x + 7.

(1.68)

12

1 Preliminaries

Equating the coeﬃcients of like terms from both sides gives A = 1,

B = 2.

(1.69)

This in turn gives u(x) = uc + up = αe2x + βe3x + x + 2, where α and β are arbitrary constants.

(1.70)

Example 1.12 Solve the following initial value problem u + 9u = 20ex,

u(0) = 3,

y (0) = 5.

(1.71)

We ﬁrst ﬁnd uc . The auxiliary equation for the related homogeneous equation is given by (1.72) r2 + 9 = 0, where we ﬁnd r = ±3i,

i2 = −1.

(1.73)

The general solution is given by u(x) = α cos(3x) + β sin(3x). (1.74) Noting that g(x) = 20ex , then a particular solution is assumed to be of the form: up = Aex . (1.75) Since this is a particular solution, then substituting up into the inhomogeneous equation leads to 10Aex = 20ex, (1.76) so that A = 2. This in turn gives the general solution u(x) = uc + up = α cos(3x) + β sin(3x) + 2ex .

(1.77) (1.78)

Since the initial conditions are given, the numerical values for α and β should be determined. Substituting the initial values into the general solution we ﬁnd α + 2 = 3,

3β + 2 = 5,

(1.79)

β = 1.

(1.80)

where we ﬁnd α = 1,

Accordingly, the particular solution is given by u(x) = cos(3x) + sin(3x) + 2ex .

Exercises 1.2.2 Find the general solution for the following second order ODEs:

(1.81)

1.2 Ordinary Diﬀerential Equations

13

1. u − 4u + 4u = 0

2. u − 2u − 3u = 0

3. u − u − 2u = 0

4. u − 2u = 0

5. u − 6u + 9u = 0

6. u + 4u = 0

Find the general solution for the following initial value problems: 7. u − 2u + 2u = 0, u(0) = 1, u (0) = 1

8. u − 6u + 9u = 0, u(0) = 1, u (0) = 4

9. u − 3u − 10u = 0, u(0) = 2, u (0) = 3 10. u + 9u = 0, u(0) = 1, u (0) = 0 11. u − 9u = 0, u(0) = 3, u (0) = 9

12. u − 9u = 0, u(0) = 1, u (0) = 0

Use the method of undetermined coeﬃcients to ﬁnd the general solution for the following second order ODEs: 13. u − u = 1

14. u + u = 3

15. u − u = 3x

16. u − u = 2 cos x

Use the method of undetermined coeﬃcients to ﬁnd the particular solution for the following initial value problems: 17. u − u = 6, u(0) = 3, u (0) = 2

18. u + u = 6ex , u(0) = 3, u (0) = 2

19. u − u = −2 sin x, u(0) = 1, u (0) = 2 20. u − 5u + 4u = −1 + 4x, u(0) = 3, u (0) = 9

1.2.3 The Series Solution Method For diﬀerential equations of any order, with constant coeﬃcients or with variable coeﬃcients, with x = 0 is an ordinary point, we can use the series solution method to determine the series solution of the diﬀerential equation. The obtained series solution may converge the exact solution if such a closed form solution exists. If an exact solution is not obtainable, we may use a truncated number of terms of the obtained series for numerical purposes. Although the series solution can be used for equations with constant coeﬃcients or with variable coeﬃcients, where x = 0 is an ordinary point, but this method is commonly used for ordinary diﬀerential equations with variable coeﬃcients where x = 0 is an ordinary point. The series solution method assumes that the solution near an ordinary point x = 0 is given by ∞ an xn , (1.82) u(x) = n=0

or by using few terms of the series u(x) = a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + a6 x6 + · · ·

(1.83)

Diﬀerentiating term by term gives u (x) = a1 + 2a2 x + 3a3 x2 + 4a4 x3 + 5a5 x4 + 6a6 x5 + · · · u (x) = 2a2 + 6a3 x + 12a4 x2 + 20a5 x3 + 30a6 x4 + · · · u (x) = 6a3 + 24a4 x + 60a5 x2 + 120a6 x3 + · · ·

(1.84)

14

1 Preliminaries

and so on. Substituting u(x) and its derivatives in the given diﬀerential equation, and equating coeﬃcients of like powers of x gives a recurrence relation that can be solved to determine the coeﬃcients an , n 0. Substituting the obtained values of an , n 0 in the series assumption (1.82) gives the series solution. As stated before, the series may converge to the exact solution. Otherwise, the obtained series can be truncated to any ﬁnite number of terms to be used for numerical calculations. The more terms we use will enhance the level of accuracy of the numerical approximation. It is interesting to point out that the series solution method can be used for homogeneous and inhomogeneous equations as well when x = 0 is an ordinary point. However, if x = 0 is a regular singular point of an ODE, then solution can be obtained by Frobenius method that will not be reviewed in this text. Moreover, the Taylor series of any elementary function involved in the diﬀerential equation should be used for equating the coeﬃcients. The series solution method will be illustrated by examining the following ordinary diﬀerential equations where x = 0 is an ordinary point. Some examples will give exact solutions, whereas others will provide series solutions that can be used for numerical purposes. Example 1.13 Find a series solution for the following second order ODE: u + u = 0.

(1.85)

Substituting the series assumption for u(x) and u (x) gives 2a2 + 6a3 x + 12a4 x2 + 20a5 x3 + 30a6 x4 + · · · +a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + · · · = 0, that can be rewritten by (a0 + 2a2 ) + (a1 + 6a3 )x + (a2 + 12a4 )x2 + (a3 + 20a5 )x3 +(a4 + 30a6 )x4 + · · · = 0.

(1.86)

(1.87)

This equation is satisﬁed only if the coeﬃcient of each power of x vanishes. This in turn gives the recurrence relation a0 + 2a2 = 0, a1 + 6a3 = 0, a2 + 12a4 = 0, .. .

a3 + 20a5 = 0,

(1.88)

By solving this recurrence relation, we obtain 1 1 a2 = − a0 , a3 = − a1 , 2! 3! (1.89) 1 1 1 1 a4 = − a2 = a0 , a5 = − a3 = a1 , · · · 12 4! 20 5! The solution in a series form is given by 1 2 1 3 1 4 1 5 u(x) = a0 1 − x + x + · · · + a1 x − x + x + · · · , (1.90) 2! 4! 3! 5!

1.2 Ordinary Diﬀerential Equations

15

and in closed form by (1.91) u(x) = a0 cos x + a1 sin x, where a0 and a1 are constants that will be determined for particular solution if initial conditions are given. Example 1.14 Find a series solution for the following second order ODE: u − xu − u = 0.

(1.92)

Substituting the series assumption for u(x), u (x) and u (x) gives 2a2 + 6a3 x + 12a4 x2 + 20a5 x3 + 30a6 x4 + 42a7 x5 + · · · −a1 x − 2a2 x2 − 3a3 x3 − 4a4 x4 − 5a5 x5 − · · · (1.93) −a0 − a1 x − a2 x2 − a3 x3 − a4 x4 − a5 x5 − · · · = 0, that can be rewritten by (−a0 + 2a2 ) + (−2a1 + 6a3 )x + (−3a2 + 12a4 )x2 + (−4a3 + 20a5 )x3 +(30a6 − 5a4 )x4 + (42a7 − 6a5 )x5 + · · · = 0.

(1.94)

This equation is satisﬁed only if the coeﬃcient of each power of x is zero. This gives the recurrence relation −a0 + 2a2 = 0, −2a1 + 6a3 = 0, −3a2 + 12a4 = 0, (1.95) −4a3 + 20a5 = 0, −5a4 + 30a6 = 0, −6a5 + 42a7 = 0, · · · where by solving this recurrence relation we ﬁnd 1 1 1 1 a3 = a1 , a4 = − a2 = a0 , a2 = a0 , 2 3 4 8 (1.96) 1 1 1 1 1 1 a5 = a3 = a1 , a6 = a4 = a0 , a7 = a5 = a1 , · · · 5 15 6 48 7 105 The solution in a series form is given by 1 1 1 u(x) = a0 1 − x2 + x4 + x6 + · · · 2 8 48 1 1 1 7 x + ··· , +a1 x + x3 + x5 + (1.97) 3 15 105 where a0 and a1 are constants, where a0 = u(0) and a1 = u (0). It is clear that a closed form solution is not obtainable. If a particular solution is required, then initial conditions u(0) and u (0) should be speciﬁed to determine the coeﬃcients a0 and a1 . Example 1.15 Find a series solution for the following second order ODE: u − xu + u = −x cos x.

(1.98)

Substituting the series assumption for u(x), u , and u (x), and using the Taylor series for cos x, we ﬁnd

16

1 Preliminaries

2a2 + 6a3 x + 12a4 x2 + 20a5 x3 + 30a6 x4 + 42a7 x5 + · · · −a1 x − 2a2 x2 − 3a3 x3 − 4a4 x4 − 5a5 x5 − · · · +a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + a5 x5 + · · · 1 1 = −x + x3 − x5 + · · · , 2! 4! that can be rewritten by

(1.99)

(a0 + 2a2 ) + 6a3 x + (−a2 + 12a4 )x2 + (−2a3 + 20a5 )x3 1 1 +(30a6 − 3a4 )x4 + (42a7 − 4a5 )x5 + · · · = −x + x3 − x5 + · · · .(1.100) 2! 4! Equating the coeﬃcient of each power of x be zero and solving the recurrence relation we obtain 1 1 a2 = − a0 , a3 = − , 2 3! 1 1 1 1 1 (1.101) a3 + = , a4 = − a2 = − a0 , a5 = 12 24 10 40 120 1 1 2 1 1 a6 = a7 = a4 = − a0 , a5 − =− ,··· 10 240 21 1008 5040 The solution in a series form is given by 1 1 1 6 x + ··· u(x) = a0 1 − x2 − x4 − (1.102) 2 24 240 1 1 1 +a1 x + − x3 + x5 − x7 + · · · , 3! 5! 7! that can be rewritten as 1 1 6 1 x + ··· u(x) = a0 1 − x2 − x4 − (1.103) 2 24 240 1 1 1 +Bx + x − x3 + x5 − x7 + · · · , 3! 5! 7! by setting B = a1 − 1. Consequently, the solution is given by 1 1 6 1 (1.104) u(x) = a0 1 − x2 + x4 − x + · · · + Bx + sin x. 2 24 240 Notice that sin x gives the particular solution that arises as an eﬀect of the inhomogeneous part.

Exercises 1.2.3 Find the series solution for the following homogeneous second order ODEs: 1. u + xu + u = 0

2. u − xu + xu = 0

3. u − (1 + x)u + u = 0

4. u − u + xu = 0

Find the series solution for the following inhomogeneous second order ODEs: 5. u − u + xu = sin x

6. u − xu + xu = ex

7. u − xu = cos x

8. u − x2 u = ln(1 − x)

1.3 Leibnitz Rule for Diﬀerentiation of Integrals

17

1.3 Leibnitz Rule for Diﬀerentiation of Integrals One of the methods that will be used to solve integral equations is the conversion of the integral equation to an equivalent diﬀerential equation. The conversion is achieved by using the well-known Leibnitz rule [4,6,7] for diﬀerentiation of integrals. Let f (x, t) be continuous and ∂f ∂t be continuous in a domain of the x-t plane that includes the rectangle a x b, t0 t t1 , and let h(x) F (x) = f (x, t)dt, (1.105) g(x)

then diﬀerentiation of the integral in (1.105) exists and is given by h(x) dh(x) dg(x) dF ∂f (x, t) = f (x, h(x)) − f (x, g(x)) + dt. F (x) = dx dx dx ∂x g(x) (1.106) If g(x) = a and h(x) = b where a and b are constants, then the Leibnitz rule (1.106) reduces to b dF ∂f (x, t) F (x) = = dt, (1.107) dx ∂x a which means that diﬀerentiation and integration can be interchanged such as b b d ext dt = text dt. (1.108) dx a a It is interested to notice that Leibnitz rule is not applicable for the Abel’s singular integral equation: x u(t) F (x) = dt, 0 < α < 1. (1.109) (x − t)α 0 The integrand in this equation does not satisfy the conditions that f (x, t) be continuous, because it is unbounded at x = t. We be continuous and ∂f ∂t illustrate the Leibnitz rule by the following examples. Example 1.16 Find F (x) for the following:

cos x

F (x) =

1 + t3 dt.

(1.110)

sin x

We can set g(x) = sin x and h(x) = cos x. It is also clear that f (x, t) is a function of t only. Using Leibnitz rule (1.106) we ﬁnd that F (x) = − sin x 1 + cos3 x − cos x 1 + sin3 x. (1.111) Example 1.17 Find F (x) for the following:

x2

(x − t) cos tdt.

F (x) = x

(1.112)

18

1 Preliminaries

We can set g(x) = x, h(x) = x2 , and f (x, t) = (x − t) cos t is a function of x and t. Using Leibnitz rule (1.106) we ﬁnd that x2 2 2 F (x) = 2x(x − x ) cos x + cos tdt, (1.113) x

or equivalently F (x) = 2x(x − x2 ) cos x2 + sin x2 − sin x.

(1.114)

Remarks In this text of integral equations, we will concern ourselves in diﬀerentiation of integrals of the form: x K(x, t)u(t)dt. (1.115) F (x) = 0

In this case, Leibnitz rule (1.106) reduces to the form: x ∂K(x, t) u(t)dt. F (x) = K(x, x)u(x) + ∂x 0 This will be illustrated by the following examples.

(1.116)

Example 1.18 Find F (x) for the following:

x

F (x) = 0

(x − t)u(t)dt.

Applying the reduced Leibnitz rule (1.116) yields x F (x) = u(t)dt.

(1.117)

(1.118)

0

Example 1.19 Find F (x) and F (x) for the following: x xtu(t)dt. F (x) =

(1.119)

0

Applying the reduced Leibnitz rule (1.116) gives x F (x) = x2 u(x) + tu(t)dt, 0

(1.120)

F (x) = x2 u (x) + 3xu(x). Example 1.20 Find F (x), F (x), F (x) and F (iv) (x) for the following integral x (x − t)3 u(t)dt. F (x) = 0

Applying the reduced Leibnitz rule (1.116) four times gives

(1.121)

1.4 Reducing Multiple Integrals to Single Integrals

F (x) = F (x) =

x 0

3(x − t)2 u(t)dt,

x

6u(t)dt,

F (x) =

19

x

6(x − t)u(t)dt,

0

(1.122)

F (iv) (x) = 6u(x).

0

Exercises 1.3 Find F (x) for the following integrals: x 2 2 1. F (x) = e−x t dt 3. F (x) =

0 x 0

sin(x2 + t2 )dt

Find F (x) for the following integrals: x (x − t)u(t)dt 5. F (x) = 0 x

7. F (x) =

0

(x − t)3 u(t)dt

2. F (x) =

x2

ln(1 + xt)dt

x x

4. F (x) =

6. F (x) =

cosh(x3 + t3 )dt

0 x

0 x

8. F (x) =

0

(x − t)2 u(t)dt (x − t)4 u(t)dt

Diﬀerentiate both sides of the following equations: x x 1 9. x3 + x6 = (4 + x − t)u(t)dt 10. 1 + xex = ex−t u(t)dt 6 0 0 x 11. 2x2 + 3x3 = (6 + 5x − 5t)u(t)dt 0

12. sinh x + ln(sin x) =

x 0

(3 + x − t)u(t)dt, 0 < x <

π 2

Diﬀerentiate the following F (x) as many times as you need to get rid of the integral sign: x x 13. F (x) = x + (x − t)u(t)dt 14. F (x) = x2 + (x − t)2 u(t)dt 15. F (x) = 1 +

0 x

0

3

(x − t) u(t)dt

x

16. F (x) = e +

0 x

0

(x − t)4 u(t)dt

Use Leibnitz rule to prove the following identities: x (x − t)n u(t)dt, show that F (n+1) = n!u(x), n 0. 17. If F (x) = 18. F (x) =

0

x

0

tn (x − t)m dt, show that F (m) =

m! n+1 x , n+1

m and n are positive integers.

1.4 Reducing Multiple Integrals to Single Integrals It will be seen later that we can convert initial value problems and other problems to integral equations. It is normal to outline the formula that will

20

1 Preliminaries

reduce multiple integrals to single integrals. We will ﬁrst show that the double integral can be reduced to a single integral by using the formula: x x1 x F (t)dtdx1 = (x − t)F (t)dt. (1.123) 0

0

0

This can be easily proved by two ways. The ﬁrst way is to set x (x − t)F (t)dt, G(x) =

(1.124)

where G(0) = 0. Diﬀerentiating both sides of (1.124) gives x G (x) = F (t)dt,

(1.125)

0

0

obtained by using the reduced Leibnitz rule. Now by integrating both sides of the last equation from 0 to x yields x x1 G(x) = F (t)dtdx1 . (1.126) 0

0

Consequently, the right side of the two equations (1.124) and (1.126) are equivalent. This completes the proof. For the second method we will use the concept of integration by parts. Recall that udv = uv − vdu, x1 (1.127) F (t)dt, u(x1 ) = 0

then we ﬁnd x x1 0

0

x x F (t)dtdx1 = x1 F (t)dt − x1 F (x1 )dx1 0 x0 x 0 =x F (t)dt − x1 F (x1 )dx1 0 0 x = (x − t)F (t)dt,

x1

(1.128)

0

obtained upon setting x1 = t. The general formula that converts multiple integrals to a single integral is given by xn−1 x x1 x 1 ··· u(xn )dxn dxn−1 · · · dx1 = (x − t)n−1 u(t)dt. (n − 1)! 0 0 0 0 (1.129) The conversion formula (1.129) is very useful and facilitates the calculation works. Moreover, this formula will be used to convert initial value problems to Volterra integral equations. Corollary As a result to (1.129) we can easily show the following corollary

1.4 Reducing Multiple Integrals to Single Integrals

0

x x 0

···

x

1 (x − t)u(t)dtdt · · · dt = n! 0 n integrals

21

0

x

(x − t)n u(t)dt

(1.130)

Example 1.21 Convert the following multiple integral to a single integral: x x1 u(x2 )dx1 dx.

(1.131)

Using the formula (1.129) we obtain x x1 x u(x2 )dx1 dx = (x − t)u(t)dt.

(1.132)

0

0

0

0

0

Example 1.22 Convert the following multiple integral to a single integral: x x1 x2 u(x3 )dx2 dx1 dx. 0

0

(1.133)

0

Using the formula (1.129) we obtain x x1 x2 1 x u(x3 )dx2 dx1 dx = (x − t)2 u(t)dt. 2! 0 0 0 0

(1.134)

Example 1.23 Convert the following multiple integral to a single integral: x x x (x − t)dtdtdt. 0

0

(1.135)

0

Using the corollary (1.130) we obtain x x x 1 x (x − t)dtdtdt = (x − t)3 u(t)dt. 3! 0 0 0 0

(1.136)

Example 1.24 Convert the following multiple integral to a single integral: x x1 (x − t)u(x1 )dtdx1 . 0

(1.137)

0

Using the corollary (1.130) we obtain x x1 1 x (x − t)u(x1 )dtdx1 = (x − t)2 u(t)dt. 2 0 0 0

(1.138)

Example 1.25 Convert the following multiple integral to a single integral: x x1 (x − t)2 u(x1 )dtdx1 . 0

0

Using the corollary (1.130) we obtain

(1.139)

22

1 Preliminaries

0

x

x1

0

(x − t)2 u(x1 )dtdx1 =

1 3

x 0

(x − t)3 u(t)dt.

(1.140)

Exercises 1.4 Prove the following: x x1 1 x (x − t)3 u(x1 )dtdx1 = (x − t)4 u(t)dt 1. 4 0 0 0 x x 1 1 x 2. (x − t)4 u(x1 )dtdx1 = (x − t)5 u(t)dt 5 0 0 0 x x x x1 1 3. (x − t)u(x1 )dtdx1 + (x − t)2 u(x1 )dtdx1 0

0

1 = 6 x 4. 0

=

0

x

0 x1

0

1 4

0

(x − t) (3 + 2(x − t))u(t)dt

u(x1 )dtdx1 +

x

0

2

x

0

x1

0

(x − t)u(x1 )dtdx1 +

x

0

x1

0

(x − t)3 u(x1 )dtdx1

(x − t)2 4 + 2(x − t) + (x − t)3 u(t)dt

1.5 Laplace Transform In this section we will review only the basic concepts of the Laplace transform method. The details can be found in any text of ordinary diﬀerential equations. The Laplace transform method is a powerful tool used for solving diﬀerential and integral equations. The Laplace transform [1–4,7] changes differential equations and integral equations to polynomial equations that can be easily solved, and hence by using the inverse Laplace transform gives the solution of the examined equation. The Laplace transform of a function f (x), deﬁned for x 0, is deﬁned by ∞ e−sx f (x)dx, (1.141) F (s) = L{f (x)} = 0

where s is real, and L is called the Laplace transform operator. The Laplace transform F (s) may fail to exist. If f (x) has inﬁnite discontinuities or if it grows up rapidly, then F (s) does not exist. Moreover, an important necessary condition for the existence of the Laplace transform F (s) is that F (s) must vanish as s approaches inﬁnity. This means that lim F (s) = 0. (1.142) s→∞

In other words, the conditions for the existence of a Laplace transform F (s) of any function f (x) are: 1. f (x) is piecewise continuous on the interval of integration 0 x < A for any positive A,

1.5 Laplace Transform

23

2. f (x) is of exponential order eax as x → ∞, i.e. |f (x)| Keax , x M , where a is real constant, and K and M are positive constants. Accordingly, the Laplace transform F (s) exists and must satisfy lim F (s) = 0. (1.143) s→∞

1.5.1 Properties of Laplace Transforms From the deﬁnition of the Laplace transform given in (1.141), we can easily derive the following properties of the Laplace transforms: 1). Constant Multiple: L{af (x)} = aL{f (x)}, a is a constant.

(1.144)

For example: L{4ex} = 4L{ex } =

4 . s−1

(1.145)

2). Linearity Property: L{af (x) + bg(x)} = aL{f (x)} + bL{g(x)}, a, b are constants. (1.146) For example: 4 6 (1.147) L{4x + 3x2 } = 4L{x} + 3L{x2 } = 2 + 3 . s s 3). Multiplication by x: d (1.148) L{xf (x)} = − L{f (x)} = −F (s). ds For example: d 1 2s d = 2 L{x sin x} = − L{sin x} = − . (1.149) 2 ds ds s + 1 (s + 1)2 To use the Laplace transform L for solving initial value problems or integral equations, we use the following table of elementary Laplace transforms as shown below: Table 1.1 Elementary Laplace Transforms f (x) c x xn eax

F (s) = L{f (x)} =

∞ 0

e−sx f (x)dx

c , s>0 s 1 , s>0 s2 n! Γ(n + 1) = , s > 0, Re n > −1 sn+1 sn+1 1 , s>a s−a

24

1 Preliminaries f (x) sin ax cos ax sin2 ax cos2 ax x sin ax x cos ax sinh ax cosh ax sinh2 ax cosh2 ax x sinh ax x cosh ax xn eax eax sin bx eax cos bx eax sinh bx eax cosh bx H(x − a) δ(x)

F (s) = L{f (x)} =

Continued ∞ 0

e

−sx

f (x)dx

a s 2 + a2 s s 2 + a2 2a2 , Re(s) > |Im(a)| s(s2 + 4a2 ) 2 2 s + 2a , Re(s) > |Im(a)| s(s2 + 4a2 ) 2as (s2 + a2 )2 s 2 − a2 (s2 + a2 )2 a , s > |a| s 2 − a2 s , s > |a| s 2 − a2 2a2 , Re(s) > |Im(a)| s(s2 − 4a2 ) s2 − 2a2 , Re(s) > |Im(a)| s(s2 − 4a2 ) 2as , s > |a| (s2 − a2 )2 2 2 s +a , s > |a| (s2 − a2 )2 n! , s > a, n is a positive integer (s − a)n+1 b , s>a (s − a)2 + b2 s−a , s>a (s − a)2 + b2 b , s>a (s − a)2 − b2 s−a , s>a (s − a)2 − b2 s−1 e−as , a 0 1

δ(x − a)

e−as , a 0

δ (x − a)

se−as , a 0

1.5 Laplace Transform

25

4). Laplace Transforms of Derivatives: L{f (x)} = sL{f (x)} − f (0), L{f (x)} = s2 L{f (x)} − sf (0) − f (0), L{f (x)} = s3 L{f (x)} − s2 f (0) − sf (0) − f (0), .. . L{f (n) (x)} = sn L{f (x)} − sn−1 f (0) − · · · − sf (n−2) (0) − f (n−1) (0). (1.150) 5). Inverse Laplace Transform If the Laplace transform of f (x) is F (s), then we say that the inverse Laplace transform of F (s) is f (x). In other words, we write (1.151) L−1 {F (s)} = f (x), where L−1 is the operator of the inverse Laplace transform. The linearity property holds also for the inverse Laplace transform. This means that L−1 {aF (s) + bG(s)} = aL−1 {F (s)} + bL−1 {G(s)} (1.152) = af (x) + bg(x). Notice that the computer symbolic systems such as Maple and Mathematica can be used to ﬁnd the Laplace transform and the inverse Laplace transform. The Laplace transform method and the inverse Laplace transform method will be illustrated by using the following examples. Example 1.26 Solve the following initial value problem: y + y = 0, y(0) = 1,

y (0) = 1.

(1.153)

By taking Laplace transforms of both sides of the ODE, we use L{y(x)} = Y (s), L{y (x)} = s2 L{y(x)} − sy(0) − y (0)

(1.154)

= s2 Y (s) − s − 1, obtained upon using the initial conditions. Substituting this into the ODE gives s 1 Y (s) = 2 + . (1.155) s + 1 s2 + 1 To determine the solution y(x) we then take the inverse Laplace transform L−1 to both sides of the last equation to ﬁnd s 1 −1 L−1 {Y (s)} = L−1 + L . (1.156) s2 + 1 s2 + 1 This in turn gives the solution by y(x) = cos x + sin x,

(1.157)

26

1 Preliminaries

obtained upon using the table of Laplace transforms. Notice that we obtained the particular solution for the diﬀerential equation. This is due to the fact that the initial conditions were used in the solution. Example 1.27 Solve the following initial value problem y − y = 0, y(0) = 2, y (0) = 1. By taking Laplace transforms of both sides of the ODE, we set L{y(x)} = Y (s), L{y (x)} = sY (s) − 2, L{y (x)} = s2 L{y(x)} − sy(0) − y (0)

(1.158)

(1.159)

= s2 Y (s) − 2s − 1, obtained by using the initial conditions. Substituting this into the ODE gives 1 1 + . (1.160) Y (s) = s−1 s To determine the solution y(x) we then the take inverse Laplace transform L−1 of both sides of the last equation to ﬁnd 1 1 + L−1 . (1.161) L−1 {Y (s)} = L−1 s−1 s This in turn gives the solution by (1.162) y(x) = ex + 1, obtained upon using the table of Laplace transforms. 6). The Convolution Theorem for Laplace Transform This is an important theorem that will be used in solving integral equations. The kernel K(x, t) of the integral equation: h(x) u(x) = f (x) + λ K(x, t)u(t)dt, (1.163) g(x)

is termed diﬀerence kernel if it depends on the diﬀerence x − t. Examples of the diﬀerence kernels are ex−t, sin(x − t), and cosh(x − t). The integral equation (1.163) can be expressed as h(x) K(x − t)u(t)dt. (1.164) u(x) = f (x) + λ g(x)

Consider two functions f1 (x) and f2 (x) that possess the conditions needed for the existence of Laplace transform for each. Let the Laplace transforms for the functions f1 (x) and f2 (x) be given by L{f1 (x)} = F1 (s), (1.165) L{f2 (x)} = F2 (s). The Laplace convolution product of these two functions is deﬁned by

1.5 Laplace Transform

27

(f1 ∗ f2 )(x) =

0

or

x

(f2 ∗ f1 )(x) =

0

f1 (x − t)f2 (t)dt,

(1.166)

f2 (x − t)f1 (t)dt.

(1.167)

x

Recall that (f1 ∗ f2 )(x) = (f2 ∗ f1 )(x).

(1.168)

We can easily show that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x f1 (x − t)f2 (t)dt = F1 (s)F2 (s). (1.169) L{(f1 ∗ f2 )(x)} = L 0

It was stated before, that this theorem will be used in the coming chapters. To illustrate the use of this theorem we examine the following example. Example 1.28 Find the Laplace transform of

x+ 0

x

(x − t)y(t)dt.

(1.170)

Notice that the kernel depends on the diﬀerence x − t. The integral includes f1 (x) = x and f2 (x) = y(x). The integral is the convolution product (f1 ∗ f2 )(x). This means that if we take Laplace transform of each term we obtain x (x − t)y(t)dt = L{x} + L{x}L{y(t)}. (1.171) L{x} + L 0

Using the table of Laplace transforms gives 1 1 + 2 Y (s). s2 s

(1.172)

Example 1.29 Solve the following integral equation x x xe = ex−t y(t)dt.

(1.173)

0

Notice that f1 (x) = ex and f2 (x) = y(x). The right hand side is the convolution product (f1 ∗ f2 )(x). This means that if we take Laplace transforms of both sides we obtain x

L{xex } = L

ex−t y(t)dt 0

= L{ex }L{y(t)}.

Using the table of Laplace transforms gives 1 1 = Y (s), (s − 1)2 s−1 that gives 1 . Y (s) = s−1

(1.174)

(1.175)

(1.176)

28

1 Preliminaries

From this we ﬁnd the solution is

1 (1.177) = ex . s−1 The Laplace transform method will be used for solving Volterra integral equations of the ﬁrst and the second type in coming chapters. y(x) = L−1

Exercises 1.5 Solve the given ODEs: 1. y + 4y = 0, y(0) = 0, y (0) = 1

2. y + 4y = 4x, y(0) = 0, y (0) = 1

3. y − y = −2x, y(0) = 0, y (0) = 1

4. y − 3y + 2y = 0, y(0) = 2, y (0) = 3

Find the Laplace transform of the following expressions: 5. x + sin x

6. ex − cos x

7. 1 + xex

8. sin x + sinh x

Find the Laplace transform of the following expressions that include convolution products: x x 9. sinh(x − t)y(t)dt 10. x2 + ex−t y(t)dt 0

0

x

11. 0

(x − t)ex−t y(t)dt

12. 1 + x −

x 0

(x − t)y(t)dt

Find the inverse Laplace transform of the following: 4 1 + 3 s2 + 1 s 1 s 15. F (s) = 2 + 2 s +1 s −4 13. F (s) =

3 1 + 2 s2 − 1 s +9 1 1 16. F (s) = − s(s2 + 1) s(s2 − 1) 14. F (s) =

1.6 Inﬁnite Geometric Series A geometric sequence is a sequence of numbers where each term after the ﬁrst is obtained by multiplying the previous term by a non-zero number r called the common ratio. In other words, a sequence is geometric if there is a ﬁxed non-zero number r such that an+1 = an r, n 1. (1.178) This means that the geometric sequence can be written in a general form as (1.179) a1 , a1 r, a1 r2 , . . . , a1 rn−1 , . . . , where a1 is the starting value of the sequence and r is the common ratio. The associated geometric series is obtained as the sum of the terms of the geometric series, and therefore given by n Sn = a1 rk = a1 + a1 r + a1 r2 + a1 r3 + a1 r4 + · · · + a1 rn . (1.180) k=0

1.6 Inﬁnite Geometric Series

29

The sum of the ﬁrst n terms of a geometric sequence is given by a1 (1 − rn ) , r = 1. (1.181) Sn = 1−r An inﬁnite geometric series converges if and only if |r| < 1. Otherwise it diverges. The sum of inﬁnite geometric series, for |r| < 1, is given by a1 lim Sn = , (1.182) n→∞ 1−r obtained from (1.181) by noting that lim rn = 0,

n→∞

|r| < 1.

(1.183)

Some of the methods that will be used in this text may give the solutions in an inﬁnite series. Some of the obtained series include inﬁnite geometric series. For this reason we will study examples of inﬁnite geometric series. Example 1.30 Find the sum of the inﬁnite geometric series 8 2 4 + ··· (1.184) 1+ + + 3 9 27 The ﬁrst value of the sequence and the common ratio are given by a1 = 1 and r = 23 respectively. The sum is therefore given by 1 S= = 3. (1.185) 1 − 23 Example 1.31 Find the sum of the inﬁnite geometric series: e−1 + e−2 + e−3 + e−4 + · · · (1.186) −1 −1 The ﬁrst term and the common ratio are a1 = e and r = e < 1. The sum is therefore given by e−1 1 . (1.187) S= = −1 1−e e−1 Example 1.32 Find the sum of the inﬁnite geometric series: n n2 n3 x+ x+ x+ x + · · · , 0 < n < 4. (1.188) 4 16 64 n It is obvious that a1 = x and r = 4 < 1. Consequently, the sum of this inﬁnite series is given by x 4 (1.189) S= n = 4 − n x, 0 < n < 4. 1− 4 Example 1.33 Find the sum of the inﬁnite geometric series:

30

1 Preliminaries

1 1 1 + + 3 + · · · , x > 1. (1.190) x x2 x 1 1 It is obvious that a1 = x and r = x < 1. The sum is therefore given by S=

1 x

1−

1 x

=

1 , x−1

x > 1.

(1.191)

Example 1.34 Find the sum of the inﬁnite geometric series: 1 1 1 1 − + − + ··· (1.192) 2 4 8 It is obvious that a1 = 1 and r = − 21 , |r| < 1. The sum is therefore given by 1 2 S= (1.193) = . 3 1 + 12 Exercises 1.6 Find the sum of the following inﬁnite series: 1 1 1 1 1 1 1 1 + + + + ··· 2. − + − + ··· 2 4 8 16 2 4 8 16 5 5 5 5 3. x + x+ x+ x + ··· 6 36 216 1296 6 6 6 6 4. sin x + sin x + sin x + sin x + · · · 7 49 343 2401 n n2 n3 6. x + x + x+ x + ··· , 0 < n < 9 5. e−2 + e−4 + e−6 + e−8 + · · · 9 81 729 π π e e2 π e3 e4 8. + 2 + 3 + 4 + ··· 7. πx + x + x + x + · · · 2 4 8 π π π π ln x ln x 2 d ln x + + + ··· = , x > 0 9. Show that dx 2 4 x e+1 1 1 1 + 2 + 3 + · · · dx = 1, x > 1 10. Show that 2 x x x 1.

References 1. C. Baker, The Numerical Treatment of Integral Equations, Oxford University Press, London, (1977). 2. M. Bocher, Integral Equations,Cambridge University Press, London, (1974). 3. L.G. Chambers, Integral Equations, A Short Course, International Textbook Company, London, (1976). 4. A.J. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 5. R.P. Kanwal and K.C. Liu, A Taylor expansion approach for solving integral equations, Int. J. Math. Educ. Sci. Technol., 20(3) (1989) 411–414.

References

31

6. R. Kress, Linear Integral Equations, Springer, Berlin, (1999). 7. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).

Chapter 2

Introductory Concepts of Integral Equations

As stated in the previous chapter, an integral equation is the equation in which the unknown function u(x) appears inside an integral sign [1–5]. The most standard type of integral equation in u(x) is of the form h(x) u(x) = f (x) + λ K(x, t)u(t)dt, (2.1) g(x)

where g(x) and h(x) are the limits of integration, λ is a constant parameter, and K(x, t) is a known function, of two variables x and t, called the kernel or the nucleus of the integral equation. The unknown function u(x) that will be determined appears inside the integral sign. In many other cases, the unknown function u(x) appears inside and outside the integral sign. The functions f (x) and K(x, t) are given in advance. It is to be noted that the limits of integration g(x) and h(x) may be both variables, constants, or mixed. Integral equations appear in many forms. Two distinct ways that depend on the limits of integration are used to characterize integral equations, namely: 1. If the limits of integration are ﬁxed, the integral equation is called a Fredholm integral equation given in the form: b u(x) = f (x) + λ K(x, t)u(t)dt, (2.2) a

where a and b are constants. 2. If at least one limit is a variable, the equation is called a Volterra integral equation given in the form: x u(x) = f (x) + λ K(x, t)u(t)dt. (2.3) a

Moreover, two other distinct kinds, that depend on the appearance of the unknown function u(x), are deﬁned as follows: 1. If the unknown function u(x) appears only under the integral sign of Fredholm or Volterra equation, the integral equation is called a ﬁrst kind Fredholm or Volterra integral equation respectively. A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

34

2 Introductory Concepts of Integral Equations

2. If the unknown function u(x) appears both inside and outside the integral sign of Fredholm or Volterra equation, the integral equation is called a second kind Fredholm or Volterra equation integral equation respectively. In all Fredholm or Volterra integral equations presented above, if f (x) is identically zero, the resulting equation: b u(x) = λ K(x, t)u(t)dt (2.4)

or

a x

K(x, t)u(t)dt

u(x) = λ

(2.5)

a

is called homogeneous Fredholm or homogeneous Volterra integral equation respectively. It is interesting to point out that any equation that includes both integrals and derivatives of the unknown function u(x) is called integro-diﬀerential equation. The Fredholm integro-diﬀerential equation is of the form: b dk u u(k) (x) = f (x) + λ K(x, t)u(t)dt, u(k) = k . (2.6) dx a However, the Volterra integro-diﬀerential equation is of the form: x dk u K(x, t)u(t)dt, u(k) = k . (2.7) u(k) (x) = f (x) + λ dx a The integro-diﬀerential equations [6] will be deﬁned and classiﬁed in this text.

2.1 Classiﬁcation of Integral Equations Integral equations appear in many types. The types depend mainly on the limits of integration and the kernel of the equation. In this text we will be concerned on the following types of integral equations.

2.1.1 Fredholm Integral Equations For Fredholm integral equations, the limits of integration are ﬁxed. Moreover, the unknown function u(x) may appear only inside integral equation in the form: b

f (x) =

K(x, t)u(t)dt.

(2.8)

a

This is called Fredholm integral equation of the ﬁrst kind. However, for Fredholm integral equations of the second kind, the unknown function u(x) appears inside and outside the integral sign. The second kind is represented by the form: b

K(x, t)u(t)dt.

u(x) = f (x) + λ a

(2.9)

2.1 Classiﬁcation of Integral Equations

35

Examples of the two kinds are given by 1 sin x − x cos x = sin(xt)u(t)dt, x2 0 and 1 1 u(x) = x + (x − t)u(t)dt, 2 −1

(2.10)

(2.11)

respectively.

2.1.2 Volterra Integral Equations In Volterra integral equations, at least one of the limits of integration is a variable. For the ﬁrst kind Volterra integral equations, the unknown function u(x) appears only inside integral sign in the form: x f (x) = K(x, t)u(t)dt. (2.12) 0

However, Volterra integral equations of the second kind, the unknown function u(x) appears inside and outside the integral sign. The second kind is represented by the form: x K(x, t)u(t)dt. (2.13) u(x) = f (x) + λ 0

Examples of the Volterra integral equations of the ﬁrst kind are x xe−x = et−x u(t)dt,

(2.14)

0

and 5x2 + x3 =

x

0

(5 + 3x − 3t)u(t)dt.

(2.15)

However, examples of the Volterra integral equations of the second kind are x u(x) = 1 − u(t)dt, (2.16) 0

and u(x) = x +

0

x

(x − t)u(t)dt.

(2.17)

2.1.3 Volterra-Fredholm Integral Equations The Volterra-Fredholm integral equations [6,7] arise from parabolic boundary value problems, from the mathematical modelling of the spatio-temporal development of an epidemic, and from various physical and biological models. The Volterra-Fredholm integral equations appear in the literature in two

36

2 Introductory Concepts of Integral Equations

forms, namely

x

u(x) = f (x) + λ1 a

and

b

K1 (x, t)u(t)dt + λ2

K2 (x, t)u(t)dt,

(2.18)

a

t

u(x, t) = f (x, t) + λ 0

Ω

F (x, t, ξ, τ, u(ξ, τ ))dξdτ, (x, t) ∈ Ω × [0, T ], (2.19)

where f (x, t) and F (x, t, ξ, τ, u(ξ, τ )) are analytic functions on D = Ω×[0, T ], and Ω is a closed subset of Rn , n = 1, 2, 3. It is interesting to note that (2.18) contains disjoint Volterra and Fredholm integral equations, whereas (2.19) contains mixed Volterra and Fredholm integral equations. Moreover, the unknown functions u(x) and u(x, t) appear inside and outside the integral signs. This is a characteristic feature of a second kind integral equation. If the unknown functions appear only inside the integral signs, the resulting equations are of ﬁrst kind, but will not be examined in this text. Examples of the two types are given by x 1 2 xu(t)dt − tu(t)dt, (2.20) u(x) = 6x + 3x + 2 − 0

0

and 1 1 u(x, t) = x + t3 + t2 − t − 2 2

t 0

1 0

(τ − ξ)dξdτ.

(2.21)

2.1.4 Singular Integral Equations Volterra integral equations of the ﬁrst kind [4,7] h(x) K(x, t)u(t)dt f (x) = λ

(2.22)

g(x)

or of the second kind

h(x)

K(x, t)u(t)dt

u(x) = f (x) +

(2.23)

g(x)

are called singular if one of the limits of integration g(x), h(x) or both are inﬁnite. Moreover, the previous two equations are called singular if the kernel K(x, t) becomes unbounded at one or more points in the interval of integration. In this text we will focus our concern on equations of the form: x 1 f (x) = u(t)dt, 0 < α < 1, (2.24) (x − t)α 0 or of the second kind: x 1 u(t)dt, 0 < α < 1. (2.25) u(x) = f (x) + α 0 (x − t) The last two standard forms are called generalized Abel’s integral equation and weakly singular integral equations respectively. For α = 12 , the equation:

2.2 Classiﬁcation of Integro-Diﬀerential Equations

37

x

1 u(t)dt (2.26) x−t 0 is called the Abel’s singular integral equation. It is to be noted that the kernel in each equation becomes inﬁnity at the upper limit t = x. Examples of Abel’s integral equation, generalized Abel’s integral equation, and the weakly singular integral equation are given by x √ 1 √ u(t)dt, (2.27) x= x−t x0 1 x3 = (2.28) 1 u(t)dt, 0 (x − t) 3 √

f (x) =

and u(x) = 1 +

√

x

x+ 0

1 1

(x − t) 3

u(t)dt,

(2.29)

respectively. Exercises 2.1 For each of the following integral equations, classify as Fredholm, Volterra, or Volterra-Fredholm integral equation and ﬁnd its kind. Classify the equation as singular or not. x x u(t)dt 2. x = (1 + x − t)u(t)dt 1. u(x) = 1 + 0

0

x

3. u(x) = e + e − 1 −

1 0

u(t)dt

π 4. x + 1 − = 2

π 2 (x − t)u(t)dt 0

x 1 1 3 1 (x − t)u(t)dt 6. u(x) = x + x3 − (x − t)u(t)dt 5. u(x) = x − − 2 3 6 0 0 x 1 1 2 1 1 (x − t)u(t)dt 8. x2 − x + = (x − t)u(t)dt 7. x3 = 6 2 3 4 0 0 x 1 3 1 9. u(x) = x + x3 − (x − t)u(t)dt − xu(t)dt 2 6 0 0 t 1 1 1 (τ − ξ)dξdτ 10. u(x, t) = x + t3 + t2 − t − 2 2 0 0 x x √ 1 1 √ u(t)dt 11. x3 + x = 12. u(x) = 1 + x2 + 5 u(t)dt x −t 0 (x − t) 6 0

2.2 Classiﬁcation of Integro-Diﬀerential Equations Integro-diﬀerential equations appear in many scientiﬁc applications, especially when we convert initial value problems or boundary value problems to integral equations. The integro-diﬀerential equations contain both integral

38

2 Introductory Concepts of Integral Equations

and diﬀerential operators. The derivatives of the unknown functions may appear to any order. In classifying integro-diﬀerential equations, we will follow the same category used before.

2.2.1 Fredholm Integro-Diﬀerential Equations Fredholm integro-diﬀerential equations appear when we convert diﬀerential equations to integral equations. The Fredholm integro-diﬀerential equation contains the unknown function u(x) and one of its derivatives u(n) (x), n 1 inside and outside the integral sign respectively. The limits of integration in this case are ﬁxed as in the Fredholm integral equations. The equation is labeled as integro-diﬀerential because it contains diﬀerential and integral operators in the same equation. It is important to note that initial conditions should be given for Fredholm integro-diﬀerential equations to obtain the particular solutions. The Fredholm integro-diﬀerential equation appears in the form: b

u(n) (x) = f (x) + λ

K(x, t)u(t)dt,

(2.30)

a

where u(n) indicates the nth derivative of u(x). Other derivatives of less order may appear with u(n) at the left side. Examples of the Fredholm integrodiﬀerential equations are given by 1 1 u (x) = 1 − x + xu(t)dt, u(0) = 0, (2.31) 3 0 and π2 xtu(t)dt, u(0) = 0, u (0) = 1. (2.32) u (x) + u (x) = x − sin x − 0

2.2.2 Volterra Integro-Diﬀerential Equations Volterra integro-diﬀerential equations appear when we convert initial value problems to integral equations. The Volterra integro-diﬀerential equation contains the unknown function u(x) and one of its derivatives u(n) (x), n 1 inside and outside the integral sign. At least one of the limits of integration in this case is a variable as in the Volterra integral equations. The equation is called integro-diﬀerential because diﬀerential and integral operators are involved in the same equation. It is important to note that initial conditions should be given for Volterra integro-diﬀerential equations to determine the particular solutions. The Volterra integro-diﬀerential equation appears in the form: x u(n) (x) = f (x) + λ K(x, t)u(t)dt, (2.33) 0

2.2 Classiﬁcation of Integro-Diﬀerential Equations

39

where u(n) indicates the nth derivative of u(x). Other derivatives of less order may appear with u(n) at the left side. Examples of the Volterra integrodiﬀerential equations are given by x 1 2 x u (x) = −1 + x − xe − tu(t)dt, u(0) = 0, (2.34) 2 0 and x

u (x)+u (x) = 1−x(sin x+cos x)−

0

tu(t)dt, u(0) = −1, u (0) = 1. (2.35)

2.2.3 Volterra-Fredholm Integro-Diﬀerential Equations The Volterra-Fredholm integro-diﬀerential equations arise in the same manner as Volterra-Fredholm integral equations with one or more of ordinary derivatives in addition to the integral operators. The Volterra-Fredholm integro-diﬀerential equations appear in the literature in two forms, namely x b (n) K1 (x, t)u(t)dt + λ2 K2 (x, t)u(t)dt, (2.36) u (x) = f (x) + λ1 a

and u

(n)

a

t (x, t) = f (x, t) + λ 0

Ω

F (x, t, ξ, τ, u(ξ, τ ))dξdτ, (x, t) ∈ Ω × [0, T ],

(2.37) where f (x, t) and F (x, t, ξ, τ, u(ξ, τ )) are analytic functions on D = Ω×[0, T ], and Ω is a closed subset of Rn , n = 1, 2, 3. It is interesting to note that (2.36) contains disjoint Volterra and Fredholm integral equations, whereas (2.37) contains mixed integrals. Other derivatives of less order may appear as well. Moreover, the unknown functions u(x) and u(x, t) appear inside and outside the integral signs. This is a characteristic feature of a second kind integral equation. If the unknown functions appear only inside the integral signs, the resulting equations are of ﬁrst kind. Initial conditions should be given to determine the particular solution. Examples of the two types are given by x 1 u (x) = 24x + x4 + 3 − (x − t)u(t)dt − tu(t)dt, u(0) = 0, (2.38) 0

and 1 1 u (x, t) = 1 + t3 + t2 − t − 2 2

0

t 0

1 0

(τ − ξ)dξdτ,

u(0, t) = t3 .

(2.39)

Exercises 2.2 For each of the following integro-diﬀerential equations, classify as Fredholm, Volterra, or Volterra-Fredholm integro-equation

40 1. u (x) = 1 +

2 Introductory Concepts of Integral Equations

2. u (x) = x +

x

xu(t)dt, u(0) = 0

0

1

0

(1 + x − t)u(t)dt, u(0) = 1, u (0) = 0

3. u (x) + u(x) = x +

tu(t)dt +

0

4. u (x) + u (x) = x + 5. u (x) + u(x) = x + 6. u (x) = 1 +

x

0

x

1

0

0

x

0

1

tu(t)dt +

u(t)dt, u(0) = 0, u (0) = 1 1

0

u(t)dt, u(0) = 0, u (0) = 1, u (0) = 1

(x − t)u(t)dt, u(0) = 1

tu(t)dt, u(0) = 0, u (0) = 1

2.3 Linearity and Homogeneity Integral equations and integro-diﬀerential equations fall into two other types of classiﬁcations according to linearity and homogeneity concepts. These two concepts play a major role in the structure of the solutions. In what follows we highlight the deﬁnitions of these concepts.

2.3.1 Linearity Concept If the exponent of the unknown function u(x) inside the integral sign is one, the integral equation or the integro-diﬀerential equation is called linear [6]. If the unknown function u(x) has exponent other than one, or if the equation contains nonlinear functions of u(x), such as eu , sinh u, cos u, ln(1 + u), the integral equation or the integro-diﬀerential equation is called nonlinear. To explain this concept, we consider the equations: x (x − t)u(t)dt, (2.40) u(x) = 1 − 0

u(x) = 1 − u(x) = 1 + 0

and

1

u (x) = 1 + 0

(x − t)u(t)dt,

(2.41)

(1 + x − t)u4 (t)dt,

(2.42)

0 x

1

xteu(t) dt,

u(0) = 1.

(2.43)

2.3 Linearity and Homogeneity

41

The ﬁrst two examples are linear Volterra and Fredholm integral equations respectively, whereas the last two are nonlinear Volterra integral equation and nonlinear Fredholm integro-diﬀerential equation respectively. It is important to point out that linear equations, except Fredholm integral equations of the ﬁrst kind, give a unique solution if such a solution exists. However, solution of nonlinear equation may not be unique. Nonlinear equations usually give more than one solution and it is not usually easy to handle. Both linear and nonlinear integral equations of any kind will be investigated in this text by using traditional and new methods.

2.3.2 Homogeneity Concept Integral equations and integro-diﬀerential equations of the second kind are classiﬁed as homogeneous or inhomogeneous, if the function f (x) in the second kind of Volterra or Fredholm integral equations or integro-diﬀerential equations is identically zero, the equation is called homogeneous. Otherwise it is called inhomogeneous. Notice that this property holds for equations of the second kind only. To clarify this concept we consider the following equations: x u(x) = sin x + xtu(t)dt, (2.44)

(2.45)

(1 + x − t)u4 (t)dt,

0

(2.46)

and

0 x

0

(x − t)2 u(t)dt,

u(x) = x + u(x) =

1

x

u (x) =

xtu(t)dt,

u (0) = 0.

u(0) = 1,

(2.47)

0

The ﬁrst two equations are inhomogeneous because f (x) = sin x and f (x) = x, whereas the last two equations are homogeneous because f (x) = 0 for each equation. We usually use speciﬁc approaches for homogeneous equations, and other methods are used for inhomogeneous equations.

Exercises 2.3 Classify the following equations as Fredholm, or Volterra, linear or nonlinear, and homogeneous or inhomogeneous x 1 1. u(x) = 1 + (x − t)2 u(t)dt 2. u(x) = cosh x + (x − t)u(t)dt 3. u(x) =

0

x 0

(2 + x − t)u(t)dt

4. u(x) = λ

0

1

−1

2

t u(t)dt

42

2 Introductory Concepts of Integral Equations

5. u(x) = 1 + x + 7. u (x) = 1 +

1 0

x

0

(x − t)

1 dt 1 + u2

(x − t)u(t)dt, u(0) = 1

6. u(x) = 1 + 8. u (x) =

x

0

1

0

u3 (t)dt

(x − t)u(t)dt, u(0) = 0

2.4 Origins of Integral Equations Integral and integro-diﬀerential equations arise in many scientiﬁc and engineering applications. Volterra integral equations and Volterra integrodiﬀerential equations can be obtained from converting initial value problems with prescribed initial values. However, Fredholm integral equations and Fredholm integro-diﬀerential equations can be derived from boundary value problems with given boundary conditions. It is important to point out that converting initial value problems to Volterra integral equations, and converting Volterra integral equations to initial value problems are commonly used in the literature. This will be explained in detail in the coming section. However, converting boundary value problems to Fredholm integral equations, and converting Fredholm integral equations to equivalent boundary value problems are rarely used. The conversion techniques will be examined and illustrated examples will be presented. In what follows we will examine the steps that we will use to obtain these integral and integro-diﬀerential equations.

2.5 Converting IVP to Volterra Integral Equation In this section, we will study the technique that will convert an initial value problem (IVP) to an equivalent Volterra integral equation and Volterra integro-diﬀerential equation as well [3]. For simplicity reasons, we will apply this process to a second order initial value problem given by y (x) + p(x)y (x) + q(x)y(x) = g(x) subject to the initial conditions:

(2.48)

y(0) = α, y (0) = β, (2.49) where α and β are constants. The functions p(x) and q(x) are analytic functions, and g(x) is continuous through the interval of discussion. To achieve our goal we ﬁrst set (2.50) y (x) = u(x), where u(x) is a continuous function. Integrating both sides of (2.50) from 0 to x yields x y (x) − y (0) = u(t)dt, (2.51) 0

2.5 Converting IVP to Volterra Integral Equation

or equivalently y (x) = β +

43

x

u(t)dt.

(2.52)

0

Integrating both sides of (2.52) from 0 to x yields x x u(t)dtdt, y(x) − y(0) = βx + 0

or equivalently

(2.53)

0

x

y(x) = α + βx + 0

(x − t)u(t)dt,

(2.54)

obtained upon using the formula that reduce double integral to a single integral that was discussed in the previous chapter. Substituting (2.50), (2.52), and (2.54) into the initial value problem (2.48) yields the Volterra integral equation: x x u(t)dt + q(x) α + βx + (x − t)u(t)dt = g(x). u(x) + p(x) β + 0

0

(2.55) The last equation can be written in the standard Volterra integral equation form: x u(x) = f (x) − K(x, t)u(t)dt, (2.56) 0

where K(x, t) = p(x) + q(x)(x − t),

(2.57)

f (x) = g(x) − [βp(x) + αq(x) + βxq(x)] .

(2.58)

and It is interesting to point out that by diﬀerentiating Volterra equation (2.56) with respect to x, using Leibnitz rule, we obtain an equivalent Volterra integro-diﬀerential equation in the form: x ∂K(x, t) u (x) + K(x, x)u(x) = f (x) − u(t)dt, u(0) = f (0). (2.59) ∂x 0 The technique presented above to convert initial value problems to equivalent Volterra integral equations can be generalized by considering the general initial value problem: (2.60) y (n) + a1 (x)y (n−1) + · · · + an−1 (x)y + an (x)y = g(x), subject to the initial conditions y(0) = c0 , y (0) = c1 , y (0) = c2 , . . . , y (n−1) (0) = cn−1 . (2.61) We assume that the functions ai (x), 1 i n are analytic at the origin, and the function g(x) is continuous through the interval of discussion. Let u(x) be a continuous function on the interval of discussion, and we consider the transformation: y (n) (x) = u(x). (2.62) Integrating both sides with respect to x gives

44

2 Introductory Concepts of Integral Equations

y

(n−1)

(x) = cn−1 +

x

u(t)dt.

(2.63)

0

Integrating again both sides with respect to x yields x x (n−2) y (x) = cn−2 + cn−1 x + u(t)dtdt 0 x 0 (x − t)u(t)dt, = cn−2 + cn−1 x +

(2.64)

0

obtained by reducing the double integral to a single integral. Proceeding as before we ﬁnd x x x 1 y (n−3) (x) = cn−3 + cn−2 x + cn−1 x2 + u(t)dtdtdt 2 0 0 0 1 1 x (x − t)2 u(t)dt. (2.65) = cn−3 + cn−2 x + cn−1 x2 + 2 2 0 Continuing the integration process leads to x n−1 ck 1 xk + (x − t)n−1 u(t)dt. (2.66) y(x) = k! (n − 1)! 0 k=0

Substituting (2.62)–(2.66) into (2.60) gives x K(x, t)u(t)dt, u(x) = f (x) −

(2.67)

0

where K(x, t) =

n k=1

and f (x) = g(x) −

n j=1

an (x − t)k−1 , (k − 1)! aj

j cn−k j−k x (j − k)!

(2.68) .

(2.69)

k=1

Notice that the Volterra integro-diﬀerential equation can be obtained by differentiating (2.67) as many times as we like, and by obtaining the initial conditions of each resulting equation. The following examples will highlight the process to convert initial value problem to an equivalent Volterra integral equation. Example 2.1 Convert the following initial value problem to an equivalent Volterra integral equation: 2 y (x) − 2xy(x) = ex , y(0) = 1. (2.70) We ﬁrst set

y (x) = u(x).

(2.71)

Integrating both sides of (2.71), using the initial condition y(0) = 1 gives x u(t)dt, (2.72) y(x) − y(0) = 0

2.5 Converting IVP to Volterra Integral Equation

or equivalently

45

x

y(x) = 1 +

u(t)dt,

(2.73)

0

Substituting (2.71) and (2.73) into (2.70) gives the equivalent Volterra integral equation: x 2 u(t)dt. (2.74) u(x) = 2x + ex + 2x 0

Example 2.2 Convert the following initial value problem to an equivalent Volterra integral equation: y (x) − y(x) = sin x, y(0) = 0, y (0) = 0. (2.75) Proceeding as before, we set y (x) = u(x). (2.76) Integrating both sides of (2.76), using the initial condition y (0) = 0 gives x u(t)dt. (2.77) y (x) = 0

Integrating (2.77) again, using the initial condition y(0) = 0 yields x x x y(x) = u(t)dtdt = (x − t)u(t)dt, 0

0

(2.78)

0

obtained upon using the rule to convert double integral to a single integral. Inserting (2.76)–(2.78) into (2.70) leads to the following Volterra integral equation: x

u(x) = sin x + 0

(x − t)u(t)dt.

(2.79)

Example 2.3 Convert the following initial value problem to an equivalent Volterra integral equation: y − y − y + y = 0, We ﬁrst set

y(0) = 1,

y (0) = 2,

y (0) = 3.

y (x) = u(x),

(2.80) (2.81)

where by integrating both sides of (2.81) and using the initial condition y (0) = 3 we obtain x u(t)dt. (2.82) y = 3 + 0

Integrating again and using the initial condition y (0) = 2 we ﬁnd x x x y (x) = 2 + 3x + u(t)dtdt = 2 + 3x + (x − t)u(t)dt. 0

0

Integrating again and using y(0) = 1 we obtain

0

(2.83)

46

2 Introductory Concepts of Integral Equations

x x x 3 2 y(x) = 1 + 2x + x + u(t)dtdtdt 2 0 x0 0 3 1 = 1 + 2x + x2 + (x − t)2 u(t)dt. (2.84) 2 2 0 Notice that in (2.83) and (2.84) the multiple integrals were reduced to single integrals as used before. Substituting (2.81) – (2.84) into (2.80) leads to the Volterra integral equation: x 1 3 2 [1 + (x − t) − (x − t)2 ]u(t)dt. (2.85) u(x) = 4 + x + x + 2 2 0 Remark We can also show that if y (iv) (x) = u(x), then x u(t)dt y (x) = y (0) + 0 x y (x) = y (0) + xy (0) + (x − t)u(t)dt 0 1 2 1 x y (x) = y (0) + xy (0) + x y (0) + (x − t)2 u(t)dt 2 2 0 1 1 1 x (x − t)3 u(t)dt. y(x) = y(0) + xy (0) + x2 y (0) + x3 y (0) + 2 6 6 0 (2.86) This process can be generalized to any derivative of a higher order. In what follows we summarize the relation between derivatives of y(x) and u(x): Table 2.1 The relation between derivatives of y(x) and u(x) y (n) (x)

Integral Equations

y (x) = u(x)

y(x) = y(0) +

y (x) = u(x)

y (x) = y (0) +

y (x) = y (0) +

x

u(t)dt

0

x

0

y (x) = y (0) + xy (0) + y(x) = y(0) + xy (0) +

u(t)dt

0

y(x) = y(0) + xy (0) +

y (x) = u(x)

x

(x − t)u(t)dt

x

u(t)dt

0

x 0

(x − t)u(t)dt

1 2 1 x y (0) + 2 2

x

0

(x − t)2 u(t)dt

2.5 Converting IVP to Volterra Integral Equation

47

2.5.1 Converting Volterra Integral Equation to IVP A well-known method for solving Volterra integral and Volterra integrodiﬀerential equation, that we will use in the forthcoming chapters, converts these equations to equivalent initial value problems. The method is achieved simply by diﬀerentiating both sides of Volterra equations [6] with respect to x as many times as we need to get rid of the integral sign and come out with a diﬀerential equation. The conversion of Volterra equations requires the use of Leibnitz rule for diﬀerentiating the integral at the right hand side. The initial conditions can be obtained by substituting x = 0 into u(x) and its derivatives. The resulting initial value problems can be solved easily by using ODEs methods that were summarized in Chapter 1. The conversion process will be illustrated by discussing the following examples. Example 2.4 Find the initial value problem equivalent to the Volterra integral equation: x x u(t)dt. (2.87) u(x) = e + 0

Diﬀerentiating both sides of (2.87) and using Leibnitz rule we ﬁnd u (x) = ex + u(x).

(2.88)

It is clear that there is no need for diﬀerentiating again because we got rid of the integral sign. To determine the initial condition, we substitute x = 0 into both sides of (2.87) to ﬁnd u(0) = 1. This in turn gives the initial value problem: u (x) − u(x) = ex , u(0) = 1. (2.89) Notice that the resulting ODE is a linear inhomogeneous equation of ﬁrst order. Example 2.5 Find the initial value problem equivalent to the Volterra integral equation: x (x − t)u(t)dt. (2.90) u(x) = x2 + 0

Diﬀerentiating both sides of (2.90) and using Leibnitz rule we ﬁnd x u (x) = 2x + u(t)dt.

(2.91)

0

To get rid of the integral sign we should diﬀerentiate (2.91) and by using Leibnitz rule we obtain the second order ODE: (2.92) u (x) = 2 + u(x). To determine the initial conditions, we substitute x = 0 into both sides of (2.90) and (2.91) to ﬁnd u(0) = 0 and u (0) = 0 respectively. This in turn gives the initial value problem: (2.93) u (x) − u(x) = 2, u(0) = 0, u (0) = 0.

48

2 Introductory Concepts of Integral Equations

Notice that the resulting ODE is a second order inhomogeneous equation. Example 2.6 Find the initial value problem equivalent to the Volterra integral equation: 1 x u(x) = sin x − (x − t)2 u(t)dt. (2.94) 2 0 Diﬀerentiating both sides of the integral equation three times to get rid of the integral sign to ﬁnd x (x − t)u(t)dt, u (x) = cos x − 0 x (2.95) u (x) = − sin x − u(t)dt, 0

u (x) = − cos x − u(x). Substituting x = 0 into (2.94) and into the ﬁrst two integro-diﬀerential equations in (2.95) gives the initial conditions: u(0) = 0, u (0) = 1, u (0) = 0. (2.96) In view of the last results, the initial value problem equivalent to the Volterra integral equation (2.94) is a third order inhomogeneous ODE given by u (x) + u(x) = − cos x,

u(0) = 0,

u (0) = 1,

u (0) = 0.

(2.97)

Exercises 2.5 Convert each of the following IVPs in 1–8 to an equivalent Volterra integral equation: 2

1. y − 4y = 0, y(0) = 1

2. y + 4xy = e−2x , y(0) = 0

3. y + 4y = 0, y(0) = 0, y (0) = 1

4. y − 6y + 8y = 1, y(0) = 1, y (0) = 1

5. y − y = 0, y(0) = 2, y (0) = y (0) = 1 6. y − 2y + y = x, y(0) = 1, y (0) = 0, y (0) = 1 7. y (iv) − y = 1, y(0) = y (0) = 0, y (0) = y (0) = 1 8. y (iv) + y + y = x, y(0) = y (0) = 1, y (0) = y (0) = 0 Convert each of the following Volterra integral equation in 9–16 to an equivalent IVP: x x 9. u(x) = x + 2 u(t)dt 10. u(x) = 1 + ex − u(t)dt 0

11. u(x) = 1 + x2 +

x

0

13. u(x) = 1 − cos x + 2 15. u(x) = 1 + 2

x 0

(x − t)u(t)dt

x

0

12. u(x) = sin x −

0 x

(x − t)u(t)dt

0

(x − t)2 u(t)dt 14. u(x) = 2 + sinh x +

(x − t)3 u(t)dt

16. u(x) = 1 + ex +

x

0

x

0

(x − t)2 u(t)dt

(1 + x − t)3 u(t)dt

2.6 Converting BVP to Fredholm Integral Equation

49

2.6 Converting BVP to Fredholm Integral Equation In this section, we will present a method that will convert a boundary value problem to an equivalent Fredholm integral equation. The method is similar to the method that was presented in the previous section for converting Volterra equation to IVP, with the exception that boundary conditions will be used instead of initial values. In this case we will determine another initial condition that is not given in the problem. The technique requires more work if compared with the initial value problems when converted to Volterra integral equations. For this reason, the technique that will be presented is rarely used. Without loss of generality, we will present two speciﬁc distinct boundary value problems (BVPs) to derive two distinct formulas that can be used for converting BVP to an equivalent Fredholm integral equation. Type I We ﬁrst consider the following boundary value problem: y (x) + g(x)y(x) = h(x), 0 < x < 1,

(2.98)

with the boundary conditions: y(0) = α,

(2.99)

y(1) = β.

We start as in the previous section and set y (x) = u(x).

(2.100)

Integrating both sides of (2.100) from 0 to x we obtain x x y (t)dt = u(t)dt,

(2.101)

0

0

that gives y (x) = y (0) +

x

u(t)dt,

(2.102)

0

where the initial condition y (0) is not given in a boundary value problem. The condition y (0) will be determined later by using the boundary condition at x = 1. Integrating both sides of (2.102) from 0 to x gives x x u(t)dt, (2.103) y(x) = y(0) + xy (0) + 0

or equivalently y(x) = α + xy (0) +

0

0

x

(x − t)u(t)dt,

(2.104)

obtained upon using the condition y(0) = α and by reducing double integral to a single integral. To determine y (0), we substitute x = 1 into both sides of (2.104) and using the boundary condition at y(1) = β we ﬁnd 1 (1 − t)u(t)dt, (2.105) y(1) = α + y (0) + 0

that gives

50

2 Introductory Concepts of Integral Equations

β = α + y (0) +

1

0

This in turn gives y (0) = (β − α) −

0

(1 − t)u(t)dt.

1

(1 − t)u(t)dt.

(2.106)

(2.107)

Substituting (2.107) into (2.104) gives x 1 x(1 − t)u(t)dt + (x − t)u(t)dt. y(x) = α + (β − α)x −

(2.108)

Substituting (2.100) and (2.108) into (2.98) yields 1 xg(x)(1 − t)u(t)dt u(x) + αg(x) + (β − α)xg(x) − 0 x g(x)(x − t)u(t)dt = h(x). +

(2.109)

From calculus we can use the formula: x 1 1 (·) = (·) + (·),

(2.110)

0

0

0

0

0

x

to carry Eq. (2.109) to

x

u(x) = h(x) − αg(x) − (β − α)xg(x) − g(x) (x − t)u(t)dt 0 x 1 (1 − t)u(t)dt + (1 − t)u(t)dt , (2.111) +xg(x) 0

that gives

u(x) = f (x) + 0

x

x

t(1 − x)g(x)u(t)dt +

1

x(1 − t)g(x)u(t)dt,

(2.112)

x

that leads to the Fredholm integral equation: 1 K(x, t)u(t)dt, u(x) = f (x) +

(2.113)

0

where f (x) = h(x) − αg(x) − (β − α)xg(x),

(2.114)

and the kernel K(x, t) is given by t(1 − x)g(x), for 0 t x, (2.115) K(x, t) = x(1 − t)g(x), for x t 1. An important conclusion can be made here. For the speciﬁc case where y(0) = y(1) = 0 which means that α = β = 0, i.e. the two boundaries of a moving string are ﬁxed, it is clear that f (x) = h(x) in this case. This means that the resulting Fredholm equation in (2.113) is homogeneous or inhomogeneous if the boundary value problem in (2.98) is homogeneous or

2.6 Converting BVP to Fredholm Integral Equation

51

inhomogeneous respectively when α = β = 0. The techniques presented above will be illustrated by the following two examples. Example 2.7 Convert the following BVP to an equivalent Fredholm integral equation: y (x) + 9y(x) = cos x, y(0) = y(1) = 0. (2.116) We can easily observe that α = β = 0, g(x) = 9 and h(x) = cos x. This in turn gives f (x) = cos x. (2.117) Substituting this into (2.113) gives the Fredholm integral equation: 1 K(x, t)u(t)dt, (2.118) u(x) = cos x + 0

where the kernel K(x, t) is given by 9t(1 − x), K(x, t) = 9x(1 − t),

for 0 t x, for x t 1.

(2.119)

Example 2.8 Convert the following BVP to an equivalent Fredholm integral equation: (2.120) y (x) + xy(x) = 0, y(0) = 0, y(1) = 2. Recall that this is a boundary value problem because the conditions are given at the boundaries x = 0 and x = 1. Moreover, the coeﬃcient of y(x) is a variable and not a constant. We can easily observe that α = 0, β = 2, g(x) = x and h(x) = 0. This in turn gives f (x) = −2x2 . (2.121) Substituting this into (2.113) gives the Fredholm integral equation: 1 K(x, t)u(t)dt, (2.122) u(x) = −2x2 + 0

where the kernel K(x, t) is given by tx(1 − x), K(x, t) = x2 (1 − t),

for 0 t x, for x t 1.

(2.123)

Type II We next consider the following boundary value problem: y (x) + g(x)y(x) = h(x), with the boundary conditions: y(0) = α1 , We again set

0 < x < 1,

y (1) = β1 .

y (x) = u(x).

(2.124) (2.125) (2.126)

52

2 Introductory Concepts of Integral Equations

Integrating both sides of (2.126) from 0 to x we obtain x x y (t)dt = u(t)dt, 0

(2.127)

0

that gives y (x) = y (0) +

x

u(t)dt,

(2.128)

0

where the initial condition y (0) is not given. The condition y (0) will be derived later by using the boundary condition at y (1) = β1 . Integrating both sides of (2.128) from 0 to x gives x x u(t)dtdt, (2.129) y(x) = y(0) + xy (0) + 0

or equivalently y(x) = α1 + xy (0) +

0

x 0

(x − t)u(t)dt,

(2.130)

obtained upon using the condition y(0) = α1 and by reducing double integral to a single integral. To determine y (0), we ﬁrst diﬀerentiate (2.130) with respect to x to get x

y (x) = y (0) +

u(t)dt,

(2.131)

0

where by substituting x = 1 into both sides of (2.131) and using the boundary condition at y (1) = β1 we ﬁnd 1 y (1) = y (0) + u(t)dt, (2.132) 0

that gives

y (0) = β1 −

1

u(t)dt.

(2.133)

0

Using (2.133) into (2.130) gives x 1 u(t)dt + (x − t)u(t)dt. y(x) = α1 + x β1 − 0

(2.134)

0

Substituting (2.126) and (2.134) into (2.124) yields 1 x u(x) + α1 g(x) + β1 xg(x) − xg(x)u(t)dt + g(x)(x − t)u(t)dt = h(x). 0

0

(2.135)

From calculus we can use the formula: x 1 1 (·) = (·) + (·), 0

0

(2.136)

x

to carry Eq. (2.135) to u(x) = h(x) − (α1 + β1 x)g(x) x +xg(x) u(t)dt + 0

1

x

x u(t)dt − g(x) (x − t)u(t)dt. (2.137) 0

2.6 Converting BVP to Fredholm Integral Equation

The last equation can be written as x u(x) = f (x) + tg(x)u(t)dt + 0

53

1

xg(x)u(t)dt,

(2.138)

x

that leads to the Fredholm integral equation: 1 K(x, t)u(t)dt, u(x) = f (x) +

(2.139)

0

where f (x) = h(x) − (α1 + β1 x)g(x), and the kernel K(x, t) is given by tg(x), K(x, t) = xg(x),

for 0 t x, for x t 1.

(2.140)

(2.141)

An important conclusion can be made here. For the speciﬁc case where y(0) = y (1) = 0 which means that α1 = β1 = 0, it is clear that f (x) = h(x) in this case. This means that the resulting Fredholm equation in (2.139) is homogeneous or inhomogeneous if the boundary value problem in (2.124) is homogeneous or inhomogeneous respectively. The second type of conversion that was presented above will be illustrated by the following two examples. Example 2.9 Convert the following BVP to an equivalent Fredholm integral equation: (2.142) y (x) + y(x) = 0, y(0) = y (1) = 0. We can easily observe that α1 = β1 = 0, g(x) = 1 and h(x) = 0. This in turn gives f (x) = 0. (2.143) Substituting this into (2.139) gives the homogeneous Fredholm integral equation: 1

u(x) =

K(x, t)u(t)dt,

(2.144)

0

where the kernel K(x, t) is given by t, K(x, t) = x,

for 0 t x, for x t 1.

(2.145)

Example 2.10 Convert the following BVP to an equivalent Fredholm integral equation: (2.146) y (x) + 2y(x) = 4, y(0) = 0, y (1) = 1. We can easily observe that α1 = 0, β1 = 1, g(x) = 2 and h(x) = 4. This in turn gives f (x) = 4 − 2x. (2.147)

54

2 Introductory Concepts of Integral Equations

Substituting this into (2.139) gives the inhomogeneous Fredholm integral equation: 1 u(x) = 4 − 2x + K(x, t)u(t)dt, (2.148) 0

where the kernel K(x, t) is given by 2t, K(x, t) = 2x,

for 0 t x, for x t 1.

(2.149)

2.6.1 Converting Fredholm Integral Equation to BVP In a previous section, we presented a technique to convert Volterra integral equation to an equivalent initial value problem. In a similar manner, we will present another technique that will convert Fredholm integral equation to an equivalent boundary value problem (BVP). In what follows we will examine two types of problems: Type I We ﬁrst consider the Fredholm integral equation given by 1 K(x, t)u(t)dt, u(x) = f (x) +

(2.150)

0

where f (x) is a given function, and the kernel K(x, t) is given by t(1 − x)g(x), for 0 t x, K(x, t) = x(1 − t)g(x), for x t 1.

(2.151)

For simplicity reasons, we may consider g(x) = λ where λ is constant. Equation (2.150) can be written as x 1 u(x) = f (x) + λ t(1 − x)u(t)dt + λ x(1 − t)u(t)dt, (2.152) 0

or equivalently u(x) = f (x) + λ(1 − x)

x

x

1

tu(t)dt + λx 0

(1 − t)u(t)dt.

(2.153)

x

Each term of the last two terms at the right side of (2.153) is a product of two functions of x. Diﬀerentiating both sides of (2.153), using the product rule of diﬀerentiation and using Leibnitz rule we obtain x tu(t) u (x) = f (x) + λx(1 − x)u(x) − λ 0 1 −λx(1 − x)u(x) + λ (1 − t)u(t)dt (2.154) x x 1 = f (x) − λ tu(t) + λ (1 − t)u(t)dt. 0

x

2.6 Converting BVP to Fredholm Integral Equation

55

To get rid of integral signs, we diﬀerentiate both sides of (2.154) again with respect to x to ﬁnd that u (x) = f (x) − λxu(x) − λ(1 − x)u(x), (2.155) that gives the ordinary diﬀerential equations: (2.156) u (x) + λu(x) = f (x). The related boundary conditions can be obtained by substituting x = 0 and x = 1 in (2.153) to ﬁnd that u(0) = f (0), u(1) = f (1). (2.157) Combining (2.156) and (2.157) gives the boundary value problem equivalent to the Fredholm equation (2.150). Recall that y (x) = u(x). Moreover, if g(x) is not a constant, we can proceed in a manner similar to the discussion presented above to obtain the boundary value problem. The technique above for type I will be explained by studying the following examples. Example 2.11 Convert the Fredholm integral equation 1 K(x, t)u(t)dt, u(x) = ex +

(2.158)

0

where the kernel K(x, t) is given by 9t(1 − x), for 0 t x, K(x, t) = 9x(1 − t), for x t 1, to an equivalent boundary value problem. The Fredholm integral equation can be written as x 1 x u(x) = e + 9(1 − x) tu(t)dt + 9x (1 − t)u(t)dt. 0

(2.160)

x

Diﬀerentiating (2.160) twice with respect to x gives x 1 tu(t)dt + 9 (1 − t)u(t)dt, u (x) = ex − 9 0

and

(2.159)

(2.161)

x

u (x) = ex − 9u(x).

(2.162)

This in turn gives the ODE: u (x) + 9u(x) = ex . The related boundary conditions are given by u(0) = f (0) = 1,

u(1) = f (1) = e,

obtained upon substituting x = 0 and x = 1 into (2.160). Example 2.12 Convert the Fredholm integral equation

(2.163) (2.164)

56

2 Introductory Concepts of Integral Equations

u(x) = x3 +

1

K(x, t)u(t)dt,

(2.165)

0

where the kernel K(x, t) is given by 4t(1 − x), for 0 t x, K(x, t) = 4x(1 − t), for x t 1, to an equivalent boundary value problem. The Fredholm integral equation can be written as x 1 3 u(x) = x + 4(1 − x) tu(t)dt + 4x (1 − t)u(t)dt. 0

(2.166)

(2.167)

x

Proceeding as before we ﬁnd u (x) = 6x − 4u(x).

(2.168)

This in turn gives the ODE: u (x) + 4u(x) = 6x,

(2.169)

with the related boundary conditions: u(0) = f (0) = 0, u(1) = f (1) = 1.

(2.170)

Type II We next consider the Fredholm integral equation given by 1 K(x, t)u(t)dt, u(x) = f (x) +

(2.171)

0

where f (x) is a given function, and the kernel K(x, t) is given by tg(x), for 0 t x, K(x, t) = xg(x), for x t 1.

(2.172)

For simplicity reasons, we again consider g(x) = λ where λ is constant. Equation (2.171) can be written as x 1 u(x) = f (x) + λ tu(t)dt + λx u(t)dt. (2.173) 0

x

Each integral at the right side of (2.173) is a product of two functions of x. Diﬀerentiating both sides of (2.173), using the product rule of diﬀerentiation and using Leibnitz rule we obtain 1 u (x) = f (x) + λ u(t)dt. (2.174) x

To get rid of integral signs, we diﬀerentiate again with respect to x to ﬁnd that (2.175) u (x) = f (x) − λu(x), that gives the ordinary diﬀerential equations. Also change equations to equation (2.176) u (x) + λu(x) = f (x).

2.6 Converting BVP to Fredholm Integral Equation

57

Notice that the boundary condition u(1) in this case cannot be obtained from (2.173). Therefore, the related boundary conditions can be obtained by substituting x = 0 and x = 1 in (2.173) and (2.174) respectively to ﬁnd that u(0) = f (0), u (1) = f (1). (2.177) Combining (2.176) and (2.177) gives the boundary value problem equivalent to the Fredholm equation (2.171). Recall that y (x) = u(x). Moreover, if g(x) is not a constant, we can proceed in a manner similar to the discussion presented above to obtain the boundary value problem. The approach presented above for type II will be illustrated by studying the following examples. Example 2.13 Convert the Fredholm integral equation: 1 K(x, t)u(t)dt, u(x) = ex +

(2.178)

0

where the kernel K(x, t) is given by 4t, for 0 t x, K(x, t) = 4x, for x t 1, to an equivalent boundary value problem. The Fredholm integral equation can be written as x 1 x u(x) = e + 4 tu(t)dt + 4x u(t)dt. 0

(2.179)

(2.180)

x

Diﬀerentiating (2.180) twice with respect to x gives 1 u(t)dt, u (x) = ex + 4

(2.181)

x

and

u (x) = ex − 4u(x).

(2.182)

This in turn gives the ODE: u (x) + 4u(x) = ex . The related boundary conditions are given by

(2.183)

u(0) = f (0) = 1, u (1) = f (1) = e, (2.184) obtained upon substituting x = 0 and x = 1 into (2.180) and (2.181) respectively. Recall that the boundary condition u(1) cannot obtained in this case. Example 2.14 Convert the Fredholm integral equation 1 K(x, t)u(t)dt, u(x) = x2 + 0

where the kernel K(x, t) is given by

(2.185)

58

2 Introductory Concepts of Integral Equations

K(x, t) =

for 0 t x,

6t,

6x, for x t 1 to an equivalent boundary value problem. The Fredholm integral equation can be written as x 1 u(x) = x2 + 6 tu(t)dt + 6x u(t)dt. 0

(2.186)

(2.187)

x

Proceeding as before we ﬁnd u (x) = 2x + 6

1

u(t)dt.

(2.188)

x

and

u (x) + 6u(x) = 2,

(2.189)

with the related boundary conditions u(0) = f (0) = 0,

u (1) = f (1) = 2.

(2.190)

Exercises 2.6 Convert each of the following BVPs in 1–8 to an equivalent Fredholm integral equation: 1. y + 4y = 0, 0 < x < 1, y(0) = y(1) = 0 2. y + xy = 0, y(0) = y(1) = 0 3. y + 2y = x, 0 < x < 1, y(0) = 1, y(1) = 0 4. y + 3xy = 4, 0 < x < 1, y(0) = 0, y(1) = 0 5. y + 4y = 0, 0 < x < 1, y(0) = 0, y (1) = 0 6. y + xy = 0, y(0) = 0, y (1) = 0 7. y + 4y = x, 0 < x < 1, y(0) = 1, y (1) = 0 8. y + 4xy = 2, 0 < x < 1, y(0) = 0, y (1) = 1 Convert each of the following Fredholm integral equation in 9–16 to an equivalent BVP: 1 3t(1 − x), for 0 t x K(x, t)u(t)dt, K(x, t) = 9. u(x) = e2x + 3x(1 − t), for x t 1 0 1 t(1 − x), for 0 t x K(x, t)u(t)dt, K(x, t) = 10. u(x) = 3x2 + x(1 − t), for x t 1 0 1 6t(1 − x), for 0 t x 11. u(x) = cos x + K(x, t)u(t)dt, K(x, t) = 6x(1 − t), for x t 1 0 1 4t(1 − x), for 0 t x K(x, t)u(t)dt, K(x, t) = 12. u(x) = sinh x + 4x(1 − t), for x t 1 0

2.7 Solution of an Integral Equation 13. u(x) = e3x + 14. u(x) = x4 +

59

1

0

K(x, t)u(t)dt, K(x, t) =

1

t,

for 0 t x

x,

for x t 1

6t,

for 0 t x

K(x, t)u(t)dt, K(x, t) =

6x, for x t 1 1 4t, for 0 t x 2 K(x, t)u(t)dt, K(x, t) = 15. u(x) = 2x + 3 + 4x, for x t 1 0 1 2t, for 0 t x 16. u(x) = ex + 1 + K(x, t)u(t)dt, K(x, t) = 2x, for xt1 0 0

2.7 Solution of an Integral Equation A solution of a diﬀerential or an integral equation arises in any of the following two types: 1). Exact solution: The solution is called exact if it can be expressed in a closed form, such as a polynomial, exponential function, trigonometric function or the combination of two or more of these elementary functions. Examples of exact solutions are as follows: u(x) = x + ex , u(x) = sin x + e2x ,

(2.191)

u(x) = 1 + cosh x + tan x, and many others. 2). Series solution: For concrete problems, sometimes we cannot obtain exact solutions. In this case we determine the solution in a series form that may converge to exact soliton if such a solution exists. Other series may not give exact solution, and in this case the obtained series can be used for numerical purposes. The more terms that we determine the higher accuracy level that we can achieve. A solution of an integral equation or integro-diﬀerential equation is a function u(x) that satisﬁes the given equation. In other words, the obtained solution u(x) must satisfy both sides of the examined equation. The following examples will be examined to explain the meaning of a solution. Example 2.15 Show that u(x) = sinh x is a solution of the Volterra integral equation: x (x − t)u(t)dt. (2.192) u(x) = x + 0

Substituting u(x) = sinh x in the right hand side (RHS) of (2.192) yields

60

2 Introductory Concepts of Integral Equations

RHS = x + 0

x

(x − t) sinh tdt

= x + (sinh t − t)|x0 = sinh x = u(x) = LHS.

(2.193)

Example 2.16 Show that u(x) = sec2 x is a solution of the Fredholm integral equation π 1 1 4 2 u(x) = − + sec x + u(t)dt. (2.194) 2 2 0 Substituting u(x) = sec2 x in the right hand side of (2.194) gives π 1 4 1 2 u(t)dt RHS = − + sec x + 2 2 0 π 1 1 = − + sec2 x + (tan t) 04 2 2 = sec2 x = u(x) = LHS. (2.195) Example 2.17 Show that u(x) = sin x is a solution of the Volterra integro-diﬀerential equation: x u (x) = 1 − u(t)dt. (2.196) 0

Proceeding as before, and using u(x) = sin x into both sides of (2.196) we ﬁnd LHS = u (x) = cos x, x x sin tdt = 1 − (− cos t) 0 = cos x. (2.197) RHS = 1 − 0

Example 2.18 Show that u(x) = x + ex is a solution of the Fredholm integro-diﬀerential equation: 1 4 x u (x) = e − x + xtu(t)dt. (2.198) 3 0 Substituting u(x) = x + ex into both sides of (2.198) we ﬁnd LHS = u (x) = ex , 1 4 x RHS = e − x + x t(t + et )dt 3 0 1 1 3 4 = ex − x + x t + tet − et = ex . 3 3 0

(2.199)

2.7 Solution of an Integral Equation

61

Example 2.19 Show that u(x) = cos x is a solution of the Volterra-Fredholm integral equation: x π2 u(x) = cos x − x + u(t)dtdt. (2.200) 0

0

Proceeding as before, and using u(x) = cos x into both sides of (2.200) we ﬁnd LHS = cos x, RHS = cos x − x +

0

x

π 2

cos tdt = cos x.

(2.201)

0

Example 2.20 Show that u(x) = ex is a solution of the Fredholm integral equation of the ﬁrst kind: 1 2 2 ex +1 − 1 = ex t u(t)dt. (2.202) x2 + 1 0 Proceeding as before, and using u(x) = ex into the right side of (2.202) we ﬁnd t=1 1 2 2 e(x +1)t ex +1 − 1 (x2 +1)t = LHS. (2.203) e dt = 2 = RHS = x +1 x2 + 1 0 t=0

Example 2.21 Show that u(x) = x is a solution of the nonlinear Fredholm integral equation π 1 1 1 u(x) = x − + dt. (2.204) 12 3 0 1 + u2 (t) Using u(x) = x into the right side of (2.204) we ﬁnd t=1 1 1 1 1 π π + + tan−1 t t=0 dt = x − RHS = x − 2 12 3 0 1 + t 12 3 π 1 π = x− + − 0 = x = LHS. (2.205) 12 3 4 Example 2.22 Find f (x) if u(x) = x2 + x3 is a solution of the Fredholm integral equation 5 1 u(x) = f (x) + (xt2 + x2 t)u(t)dt. (2.206) 2 −1 Using u(x) = x2 + x3 into both sides of (2.206) we ﬁnd LHS = x2 + x3 , 5 1 (xt2 + x2 t)u(t)dt = f (x) + x2 + x. RHS = f (x) + 2 −1 Equating the left and right sides gives

(2.207)

62

2 Introductory Concepts of Integral Equations

f (x) = x3 − x.

(2.208)

Exercises 2.7 In Exercises 1–4, show that the given function u(x) is a solution of the corresponding Fredholm integral equation: π 1 2 sin xu(t)dt, u(x) = sin x + cos x 1. u(x) = cos x + 2 0 1 1 5 1 2. u(x) = e2x+ 3 − e2x− 3 t u(t)dt, u(x) = e2x 3 0 1 3. u(x) = x + (x4 − t4 )u(t)dt, − 1 x 1, u(x) = x −1

4. u(x) = x + (1 − x)ex +

1 0

x2 et(x−1) u(t)dt, u(x) = ex

In Exercises 5–8, show that the given function u(x) is a solution of the corresponding Volterra integral equation: 1 x 5. u(x) = 1 + u(t)dt, u(x) = e2x 2 0 x 6. u(x) = 4x + sin x + 2x2 − cos x + 1 − u(t)dt, u(x) = 4x + sin x 1 7. u(x) = 1 − x2 − 2

0

x 0

(x − t)u(t)dt, u(x) = 2 cos x − 1

8. u(x) = 1 + 2x + sin x + x2 − cos x −

x 0

u(t)dt, u(x) = 2x + sin x

In Exercises 9–12, show that the given function u(x) is a solution of the corresponding Fredholm integro-diﬀerential equation: 1 1 xu(t)dt, u(0) = 0, u(x) = xex 9. u (x) = xex + ex − x + 2 0 1 10. u (x) = ex + (e − 1) − u(t)dt, u(0) = 1, u(x) = ex 11. u (x) = 1 − sin x −

0

12. u (x) = 1 + sin x −

π 2

0

π 2

0

tu(t)dt, u(0) = 0, u (0) = 1, u(x) = sin x (x − t)u(t)dt,

u(0) = 1, u (0) = 0, u (0) = −1, u(x) = cos x In Exercises 13–16, show that the given function u(x) is a solution of the corresponding Volterra integro-diﬀerential equation: x 13. u (x) = 2 + x + x2 − u(t)dt, u(0) = 1, u(x) = 1 + 2x 0

14. u (x) = x cos x − 2 sin x + 15. u (x) = 1 +

x 0

x 0

tu(t)dt, u(0) = 0, u (0) = 1, u(x) = sin x

(x − t)u(t)dt, u(0) = 1, u (0) = 0, u(x) = cosh x

References

63

16. u (x) = 1 − xe−x −

x

0

tu(t)dt, u(0) = 1, u (0) = −1, u(x) = e−x

In Exercises 17–24, ﬁnd the unknown if the solution of each equation is given: x 17. If u(x) = e4x is a solution of u(x) = f (x) + 16 (x − t)u(t)dt, ﬁnd f (x) 0

18. If u(x) = e2x is a solution of u(x) = e2x − α(e2 + 1)x + 19. If u(x) = sin x is a solution of u(x) = f (x) + sin x − 2

20. If u(x) = e−x is a solution of u(x) = 1 − α 21. If u(x) = ex is a solution of u(x) = f (x) +

0

24. If u(x) = 6x is a solution of u(x) = f (x) + x

1

0

x 0

xtu(t)dt, ﬁnd α

xu(t)dt, ﬁnd f (x)

(2u2 (t) + u(t))dt, ﬁnd f (x)

x

0

0

π 2

u2 (t)dtdt, ﬁnd f (x)

23. If u(x) = 2 + 12x2 is a solution of u (x) = f (x) + 20x − ﬁnd f (x)

0

tu(t)dt, ﬁnd α

4 π

22. If u(x) = sin x is a solution of u(x) = f (x) +

0

1

x

0 x

π 2

x

0

1

0

(x − t)u(t)dtdt,

(1 − t)u(t)dt−

(x − t)u(t)dtdt, ﬁnd f (x)

References 1. C.D. Green, Integral Equations Methods, Barnes and Noble, New York, (1969). 2. H. Hochstadt, Integral Equations,Wiley, New York, (1973). 3. A. Jerri, Introduction to Integral Equations with Applications,Wiley, New York, (1999). 4. R. Kanwal, Linear Integral Equations, Birkhauser, Boston, (1997). 5. R. Kress, Linear Integral Equations, Springer, Berlin, (1999). 6. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 7. K. Maleknejad and Y. Mahmoudi, Taylor polynomial solution of highorder nonlinear Volterra-Fredholm integro-diﬀerential equations, Appl. Math. Comput. 145 (2003) 641–653. 8. A.M. Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. Comput., 127 (2002) 405–414.

Chapter 3

Volterra Integral Equations

3.1 Introduction It was stated in Chapter 2 that Volterra integral equations arise in many scientiﬁc applications such as the population dynamics, spread of epidemics, and semi-conductor devices. It was also shown that Volterra integral equations can be derived from initial value problems. Volterra started working on integral equations in 1884, but his serious study began in 1896. The name integral equation was given by du Bois-Reymond in 1888. However, the name Volterra integral equation was ﬁrst coined by Lalesco in 1908. Abel considered the problem of determining the equation of a curve in a vertical plane. In this problem, the time taken by a mass point to slide under the inﬂuence of gravity along this curve, from a given positive height, to the horizontal axis is equal to a prescribed function of the height. Abel derived the singular Abel’s integral equation, a speciﬁc kind of Volterra integral equation, that will be studied in a forthcoming chapter. Volterra integral equations, of the ﬁrst kind or the second kind, are characterized by a variable upper limit of integration [1]. For the ﬁrst kind Volterra integral equations, the unknown function u(x) occurs only under the integral sign in the form: x

f (x) =

K(x, t)u(t)dt.

(3.1)

0

However, Volterra integral equations of the second kind, the unknown function u(x) occurs inside and outside the integral sign. The second kind is represented in the form: x u(x) = f (x) + λ K(x, t)u(t)dt. (3.2) 0

The kernel K(x, t) and the function f (x) are given real-valued functions, and λ is a parameter [2–4]. A variety of analytic and numerical methods, such as successive approximations method, Laplace transform method, spline collocation method, A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

66

3 Volterra Integral Equations

Runge-Kutta method, and others have been used to handle Volterra integral equations. In this text we will apply the recently developed methods, namely, the Adomian decomposition method (ADM), the modiﬁed decomposition method (mADM), and the variational iteration method (VIM) to handle Volterra integral equations. Some of the traditional methods, namely, successive approximations method, series solution method, and the Laplace transform method will be employed as well. The emphasis in this text will be on the use of these methods and approaches rather than proving theoretical concepts of convergence and existence. The theorems of uniqueness, existence, and convergence are important and can be found in the literature. The concern will be on the determination of the solution u(x) of the Volterra integral equation of ﬁrst and second kind.

3.2 Volterra Integral Equations of the Second Kind We will ﬁrst study Volterra integral equations of the second kind given by x u(x) = f (x) + λ K(x, t)u(t)dt. (3.3) 0

The unknown function u(x), that will be determined, occurs inside and outside the integral sign. The kernel K(x, t) and the function f (x) are given real-valued functions, and λ is a parameter. In what follows we will present the methods, new and traditional, that will be used.

3.2.1 The Adomian Decomposition Method The Adomian decomposition method (ADM) was introduced and developed by George Adomian in [5–7] and is well addressed in many references. A considerable amount of research work has been invested recently in applying this method to a wide class of linear and nonlinear ordinary diﬀerential equations, partial diﬀerential equations and integral equations as well. The Adomian decomposition method consists of decomposing the unknown function u(x) of any equation into a sum of an inﬁnite number of components deﬁned by the decomposition series ∞ u(x) = un (x), (3.4) n=0

or equivalently u(x) = u0 (x) + u1 (x) + u2 (x) + · · · ,

(3.5)

where the components un (x), n 0 are to be determined in a recursive manner. The decomposition method concerns itself with ﬁnding the components

3.2 Volterra Integral Equations of the Second Kind

67

u0 , u1 , u2 , . . . individually. As will be seen through the text, the determination of these components can be achieved in an easy way through a recurrence relation that usually involves simple integrals that can be easily evaluated. To establish the recurrence relation, we substitute (3.4) into the Volterra integral equation (3.3) to obtain ∞ x ∞ un (x) = f (x) + λ K(x, t) un (t) dt, (3.6) 0

n=0

n=0

or equivalently

u0 (x) + u1 (x) + u2 (x) + · · · = f (x) + λ

x 0

K(x, t) [u0 (t) + u1 (t) + · · · ] dt.

(3.7) The zeroth component u0 (x) is identiﬁed by all terms that are not included under the integral sign. Consequently, the components uj (x), j 1 of the unknown function u(x) are completely determined by setting the recurrence relation: u0 (x) = f (x), x (3.8) K(x, t)un (t)dt, n 0, un+1 (x) = λ 0

that is equivalent to u0 (x) = f (x), u2 (x) = λ

0

u1 (x) = λ

x

K(x, t)u1 (t)dt,

u3 (x) = λ

x

K(x, t)u0 (t)dt,

0

(3.9)

x 0

K(x, t)u2 (t)dt,

and so on for other components. In view of (3.9), the components u0 (x), u1 (x), u2 (x), u3 (x), . . . are completely determined. As a result, the solution u(x) of the Volterra integral equation (3.3) in a series form is readily obtained by using the series assumption in (3.4). It is clearly seen that the decomposition method converted the integral equation into an elegant determination of computable components. It was formally shown by many researchers that if an exact solution exists for the problem, then the obtained series converges very rapidly to that solution. The convergence concept of the decomposition series was thoroughly investigated by many researchers to conﬁrm the rapid convergence of the resulting series. However, for concrete problems, where a closed form solution is not obtainable, a truncated number of terms is usually used for numerical purposes. The more components we use the higher accuracy we obtain. Example 3.1 Solve the following Volterra integral equation: x u(t)dt. u(x) = 1 − 0

(3.10)

68

3 Volterra Integral Equations

We notice that f (x) = 1, λ = −1, K(x, t) = 1. Recall that the solution u(x) is assumed to have a series form given in (3.4). Substituting the decomposition series (3.4) into both sides of (3.10) gives x ∞ ∞ un (x) = 1 − un (t)dt, (3.11) 0

n=0

or equivalently

u0 (x) + u1 (x) + u2 (x) + · · · = 1 −

n=0

x 0

[u0 (t) + u1 (t) + u2 (t) + · · · ] dt. (3.12)

We identify the zeroth component by all terms that are not included under the integral sign. Therefore, we obtain the following recurrence relation: u0 (x) = 1, x (3.13) uk (t)dt, k 0, uk+1 (x) = − 0

so that u0 (x) = 1, u1 (x) = − u2 (x) = − u3 (x) = − u4 (x) = −

x

u0 (t)dt = −

0

u1 (t)dt = −

0

u2 (t)dt = −

0

0

0

u3 (t)dt = −

x

0

x

x

(−t)dt =

x

1dt = −x,

0

x

x

0

1 2 x , 2!

(3.14)

1 1 2 t dt = − x3 , 2! 3!

x

−

1 3 1 t dt = − x4 , 3! 4!

and so on. Using (3.4) gives the series solution: 1 1 1 u(x) = 1 − x + x2 − x3 + x4 + · · · , 2! 3! 4! that converges to the closed form solution: u(x) = e−x .

(3.15) (3.16)

Example 3.2 Solve the following Volterra integral equation: x (t − x)u(t)dt. u(x) = 1 + 0

(3.17)

We notice that f (x) = 1, λ = 1, K(x, t) = t − x. Substituting the decomposition series (3.4) into both sides of (3.17) gives x ∞ ∞ un (x) = 1 + (t − x)un (t)dt, (3.18) n=0

or equivalently

0

n=0

3.2 Volterra Integral Equations of the Second Kind

u0 (x) + u1 (x) + u2 (x) + · · · = 1 +

69

x 0

(t − x) [u0 (t) + u1 (t) + u2 (t) + · · · ] dt. (3.19)

Proceeding as before we set the following recurrence relation: u0 (x) = 1, x uk+1 (x) = (t − x)uk (t)dt, k 0,

(3.20)

0

that gives u0 (x) = 1, x x 1 (t − x)u0 (t)dt = (t − x)dt = − x2 , u1 (x) = 2! 0 0 x 1 x 1 (t − x)u1 (t)dt = − (t − x)t2 dt = x4 , u2 (x) = 2! 0 4! 0 x 1 x 1 u3 (x) = (t − x)u2 (t)dt = (t − x)t4 dt = − x6 , 4! 0 6! 0 x x 1 1 (t − x)u3 (t)dt = − (t − x)t6 dt = x8 , u4 (x) = 6! 8! 0 0 and so on. The solution in a series form is given by 1 1 1 1 u(x) = 1 − x2 + x4 − x6 + x8 + · · · , 2! 4! 6! 8! and in a closed form by u(x) = cos x,

(3.21)

(3.22) (3.23)

obtained upon using the Taylor expansion for cos x. Example 3.3 Solve the following Volterra integral equation: x 1 u(x) = 1 − x − x2 − (t − x)u(t)dt. (3.24) 2 0 Notice that f (x) = 1 − x − 12 x2 , λ = −1, K(x, t) = t − x. Substituting the decomposition series (3.4) into both sides of (3.24) gives x ∞ ∞ 1 un(x) = 1 − x − x2 − (t − x)un (t)dt, (3.25) 2 0 n=0 n=0 or equivalently 1 u0 (x) + u1 (x) + u2 (x) + · · · = 1 − x − x2 − 2

0

x

(t − x) [u0 (t) + u1 (t) + · · · ] dt.

This allows us to set the following recurrence relation: 1 u0 (x) = 1 − x − x2 , 2 x (t − x)uk (t)dt, k 0, uk+1 (x) = − 0

(3.26)

(3.27)

70

3 Volterra Integral Equations

that gives 1 u0 (x) = 1 − x − x2 , 2 x 1 1 1 u1 (x) = − (t − x)u0 (t)dt = x2 − x3 − x4 , 2! 3! 4! 0 x 1 1 1 (t − x)u1 (t)dt = x4 − x5 − x6 , u2 (x) = − 4! 5! 6! 0 x 1 1 1 u3 (x) = − (t − x)u2 (t)dt = x6 − x7 − x8 , 6! 7! 8! 0 and so on. The solution in a series form is given by 1 1 1 u(x) = 1 − (x + x3 + x5 + x7 + · · · ), 3! 5! 7! and in a closed form by u(x) = 1 − sinh x,

(3.28)

(3.29) (3.30)

obtained upon using the Taylor expansion for sinh x. Example 3.4 We consider here the Volterra integral equation: x tu(t)dt. u(x) = 5x3 − x5 +

(3.31)

0

Identifying the zeroth component u0 (x) by the ﬁrst two terms that are not included under the integral sign, and using the ADM we set the recurrence relation as u0 (x) = 5x3 − x5 , x (3.32) tuk (t)dt, k 0. uk+1 (x) = 0

This in turn gives u0 (x) = 5x3 − x5 , x 1 tu0 (t)dt = x5 − x7 , u1 (x) = 7 0 x (3.33) 1 1 u2 (x) = tu1 (t)dt = x7 − x9 , 7 63 0 x 1 9 1 11 tu1 (t)dt = u3 (x) = x − x , 63 693 0 The solution in a series form is given by 3

1 7 1 9 1 11 1 7 1 5 5 u(x) = 5x − x + x − x + x − x + ( x9 − x )+· · · . 7 7 63 63 693 (3.34) We can easily notice the appearance of identical terms with opposite signs. Such terms are called noise terms that will be discussed later. Canceling the identical terms with opposite signs gives the exact solution

3.2 Volterra Integral Equations of the Second Kind

u(x) = 5x3 ,

71

(3.35)

that satisﬁes the Volterra integral equation (3.31). Example 3.5 We now consider the Volterra integral equation: x 1 1 u(t)dt. (3.36) u(x) = x + x4 + x2 + x5 − 2 5 0 Identifying the zeroth component u0 (x) by the ﬁrst four terms that are not included under the integral sign, and using the ADM we set the recurrence relation as 1 1 u0 (x) = x + x4 + x2 + x5 , 2 5 x (3.37) uk+1 (x) = − uk (t)dt, k 0. 0

This in turn gives 1 1 u0 (x) = x + x4 + x2 + x5 , 2 5 x 1 1 1 1 u1 (x) = − u0 (t)dt = − x2 − x5 − x3 − x6 , 2 5 6 30 0 x (3.38) 1 1 1 1 7 x , u1 (t)dt = x3 + x6 + x4 + u2 (x) = − 6 30 24 210 0 x 1 4 1 7 u2 (t)dt = − x − u3 (x) = − x + ··· . 24 210 0 The solution in a series form is given by 1 2 1 5 1 3 1 1 1 u(x) = x + x4 + x2 + x5 − x + x + x + x6 2 5 2 5 6 30 1 6 1 4 1 7 1 7 1 3 1 4 + x + x + x + x − x + x + ··· + ··· . 6 30 24 210 24 210 (3.39) We can easily notice the appearance of identical terms with opposite signs. This phenomenon of such terms is called noise terms phenomenon that will be presented later. Canceling the identical terms with opposite terms gives the exact solution u(x) = x + x4 . (3.40) Example 3.6 We ﬁnally solve the Volterra integral equation: 1 x 3 xt u(t)dt. u(x) = 2 + 3 0 Proceeding as before we set the recurrence relation

(3.41)

72

3 Volterra Integral Equations

u0 (x) = 2,

uk+1 (x) =

1 3

x 0

xt3 uk (t)dt,

k 0.

(3.42)

This in turn gives u0 (x) = 2, 1 x 3 1 u1 (x) = xt u0 (t)dt = x5 , 3 0 6 x 1 1 10 x , xt3 u1 (t)dt = u2 (x) = 3 0 162 (3.43) 1 x 3 1 15 xt u2 (t)dt = u3 (x) = x , 3 0 6804 1 x 3 1 u4 (x) = x20 , xt u3 (t)dt = 3 0 387828 and so on. The solution in a series form is given by 1 1 1 1 x15 + x20 + · · · . (3.44) x10 + u(x) = 2 + x5 + 3 4 5 6 6·3 6 · 3 · 14 6 · 3 · 14 · 19 It seems that an exact solution is not obtainable. The obtained series solution can be used for numerical purposes. The more components that we determine the higher accuracy level that we can achieve.

Exercises 3.2.1 In Exercises 1–26, solve the following Volterra integral equations by using the Adomian decomposition method: x x u(t)dt 2. u(x) = 6x − x3 + (x − t)u(t)dt 1. u(x) = 6x − 3x2 + 0

x 1 3. u(x) = 1 − x2 + u(t)dt 2 0 x 5. u(x) = 1 + x + (x − t)u(t)dt 0 x

0

x 2 4. u(x) = x − x3 − 2 u(t)dt 3 0 x 6. u(x) = 1 − x + (x − t)u(t)dt

7. u(x) = 1 + x − 9. u(x) = 1 −

x 0

11. u(x) = x − 13. u(x) = 1 +

0

0 x

8. u(x) = 1 − x − 10. u(x) = 1 +

0

12. u(x) = x +

u(t)dt

0

17. u(x) = 1 − x2 −

14. u(x) = 1 −

0 x

tu(t)dt

x

0

(x − t)u(t)dt

16. u(x) = 1 − 2

(x − t)u(t)dt

(x − t)u(t)dt

x

(x − t)u(t)dt u(t)dt

0

x

0

x

(x − t)u(t)dt

0

15. u(x) = 1 + 2

(x − t)u(t)dt

(x − t)u(t)dt x

0 x

x 0

tu(t)dt

18. u(x) = −2 + 3x − x2 −

x

0

(x − t)u(t)dt

3.2 Volterra Integral Equations of the Second Kind x 19. u(x) = x2 + (x − t)u(t)dt 20. u(x) = −2 + 2x + x2 + 0

21. u(x) = 1 + 2x + 4

x

0

(x − t)u(t)dt

22. u(x) = 5 + 2x2 −

73 x

0

x 0

(x − t)u(t)dt

(x − t)u(t)dt

1 2 1 x x + (x − t)2 u(t)dt 2 2 0 1 x 1 (x − t)3 u(t)dt 24. u(x) = 1 − x2 + 2 6 0 1 1 x 25. u(x) = 1 + x + (x − t + 1)u(t)dt 2 2 0 x 26. u(x) = 1 + x2 − (x − t + 1)2 u(t)dt 23. u(x) = 1 + x +

0

In Exercises 27–30, use the Adomian decomposition method to ﬁnd the series solution 1 x 2 1 x 27. u(x) = 3 + xt u(t)dt 28. u(x) = 3 + (x + t2 )u(t)dt 4 0 4 0 1 x 2 1 x 2 29. u(x) = 1 + (x − t2 )u(t)dt 30. u(x) = 1 + x u(t)dt 2 0 2 0

3.2.2 The Modiﬁed Decomposition Method As shown before, the Adomian decomposition method provides the solution in an inﬁnite series of components. The components uj , j 0 are easily computed if the inhomogeneous term f (x) in the Volterra integral equation: x K(x, t)u(t)dt (3.45) u(x) = f (x) + λ 0

consists of a polynomial. However, if the function f (x) consists of a combination of two or more of polynomials, trigonometric functions, hyperbolic functions, and others, the evaluation of the components uj , j 0 requires cumbersome work. A reliable modiﬁcation of the Adomian decomposition method was developed by Wazwaz and presented in [7–9]. The modiﬁed decomposition method will facilitate the computational process and further accelerate the convergence of the series solution. The modiﬁed decomposition method will be applied, wherever it is appropriate, to all integral equations and diﬀerential equations of any order. It is interesting to note that the modiﬁed decomposition method depends mainly on splitting the function f (x) into two parts, therefore it cannot be used if the function f (x) consists of only one term. The modiﬁed decomposition method will be outlined and employed in this section and in other chapters as well. To give a clear description of the technique, we recall that the standard Adomian decomposition method admits the use of the recurrence relation:

74

3 Volterra Integral Equations

u0 (x) = f (x), x K(x, t)uk (t)dt, k 0, uk+1 (x) = λ

(3.46)

0

where the solution u(x) is expressed by an inﬁnite sum of components deﬁned before by ∞ un (x). (3.47) u(x) = n=0

In view of (3.46), the components un (x), n 0 can be easily evaluated. The modiﬁed decomposition method [7–9] introduces a slight variation to the recurrence relation (3.46) that will lead to the determination of the components of u(x) in an easier and faster manner. For many cases, the function f (x) can be set as the sum of two partial functions, namely f1 (x) and f2 (x). In other words, we can set (3.48) f (x) = f1 (x) + f2 (x). In view of (3.48), we introduce a qualitative change in the formation of the recurrence relation (3.46). To minimize the size of calculations, we identify the zeroth component u0 (x) by one part of f (x), namely f1 (x) or f2 (x). The other part of f (x) can be added to the component u1 (x) among other terms. In other words, the modiﬁed decomposition method introduces the modiﬁed recurrence relation: u0 (x) = f1 (x), x K(x, t)u0 (t)dt, u1 (x) = f2 (x) + λ (3.49) 0 x uk+1 (x) = λ K(x, t)uk (t)dt, k 1. 0

This shows that the diﬀerence between the standard recurrence relation (3.46) and the modiﬁed recurrence relation (3.49) rests only in the formation of the ﬁrst two components u0 (x) and u1 (x) only. The other components uj , j 2 remain the same in the two recurrence relations. Although this variation in the formation of u0 (x) and u1 (x) is slight, however it plays a major role in accelerating the convergence of the solution and in minimizing the size of computational work. Moreover, reducing the number of terms in f1 (x) aﬀects not only the component u1 (x), but also the other components as well. This result was conﬁrmed by several research works. Two important remarks related to the modiﬁed method [7–9] can be made here. First, by proper selection of the functions f1 (x) and f2 (x), the exact solution u(x) may be obtained by using very few iterations, and sometimes by evaluating only two components. The success of this modiﬁcation depends only on the proper choice of f1 (x) and f2 (x), and this can be made through trials only. A rule that may help for the proper choice of f1 (x) and f2 (x) could not be found yet. Second, if f (x) consists of one term only, the standard decomposition method can be used in this case.

3.2 Volterra Integral Equations of the Second Kind

75

It is worth mentioning that the modiﬁed decomposition method will be used for Volterra and Fredholm integral equations, linear and nonlinear equations. The modiﬁed decomposition method will be illustrated by discussing the following examples. Example 3.7 Solve the Volterra integral equation by using the modiﬁed decomposition method: x u(x) = sin x + (e − ecos x ) − ecos t u(t)dt. (3.50) 0

We ﬁrst split f (x) given by f (x) = sin x + (e − ecos x ),

(3.51)

f1 (x) = sin x, f2 (x) = e − ecos x .

(3.52)

into two parts, namely

We next use the modiﬁed recurrence formula (3.49) to obtain u0 (x) = f1 (x) = sin x, x cos x u1 (x) = (e − e )− ecos t u0 (t)dt = 0, 0

uk+1 (x) = −

(3.53)

x

K(x, t)uk (t)dt = 0,

0

k 1.

It is obvious that each component of uj , j 1 is zero. This in turn gives the exact solution by u(x) = sin x. (3.54) Example 3.8 Solve the Volterra integral equation by using the modiﬁed decomposition method: x π sec x (3.55) u(x) = sec x tan x + (e − e) − esec t u(t)dt, x < . 2 0 Proceeding as before we split f (x) into two parts f1 (x) = sec x tan x,

f2 (x) = esec x − e.

We next use the modiﬁed recurrence formula (3.49) to obtain u0 (x) = f1 (x) = sec x tan x, x esec t u0 (t)dt = 0, u1 (x) = (esec x − e) − 0

uk+1 (x) = −

(3.56)

(3.57)

x 0

K(x, t)uk (t)dt = 0,

k 1.

It is obvious that each component of uj , j 1 is zero. This in turn gives the exact solution by

76

3 Volterra Integral Equations

u(x) = sec x tan x.

(3.58)

Example 3.9 Solve the Volterra integral equation by using the modiﬁed decomposition method: x u(x) = 2x + sin x + x2 − cos x + 1 − u(t)dt. (3.59) 0

The function f (x) consists of ﬁve terms. By trial we divide f (x) given by f (x) = 2x + sin x + x2 − cos x + 1, into two parts, ﬁrst two terms and the next three terms to ﬁnd f1 (x) = 2x + sin x, f2 (x) = x2 − cos x + 1. We next use the modiﬁed recurrence formula (3.49) to obtain u0 (x) = 2x + sin x, x u0 (t)dt = 0, u1 (x) = x2 − cos x + 1 − 0

uk+1 (x) = −

(3.60)

(3.61)

(3.62)

x 0

K(x, t)uk (t)dt = 0,

k 1.

It is obvious that each component of uj , j 1 is zero. The exact solution is given by u(x) = 2x + sin x. (3.63) Example 3.10 Solve the Volterra integral equation by using the modiﬁed decomposition method: x 1 u(x) = 1 + x2 + cos x − x − x3 − sin x + u(t)dt. (3.64) 3 0 The function f (x) consists of six terms. By trial we split f (x) given by 1 (3.65) f (x) = 1 + x2 + cos x − x − x3 − sin x, 3 into two parts, the ﬁrst three terms and the next three terms, hence we set f1 (x) = 1 + x2 + cos x, (3.66) 1 f2 (x) = −(x + x3 + sin x). 3 Using the modiﬁed recurrence formula (3.49) gives u0 (x) = 1 + x2 + cos x, x 1 3 u1 (x) = −(x + x + sin x) + u0 (t)dt = 0, 3 (3.67) 0 x K(x, t)uk (t)dt = 0, k 1. uk+1 (x) = 0

3.2 Volterra Integral Equations of the Second Kind

77

As a result, the exact solution is given by u(x) = 1 + x2 + cos x.

(3.68)

Exercises 3.2.2 Use the modiﬁed decomposition method to solve the following Volterra integral equations: x u(t)dt 1. u(x) = cos x + sin x − 0

2. u(x) = sinh x + cosh x − 1 − 3. u(x) = 2x = 3x2 + (ex

2

+x3

3

4. u(x) = 3x2 + (1 − e−x ) − 2

5. u(x) = 2x − (1 − e−x ) +

x

0

u(t)dt

− 1) −

x

0 x 0

3

2

et

0

e−x

e−x

x

+t3

+t2

2

+t3

u(t)dt

u(t)dt

u(t)dt

x 2 x xtu(t)dt (1 − e−x ) − 2 0 x xu(t)dt 7. u(x) = cosh x + x sinh x − 2

6. u(x) = e−x −

x

x

8. u(x) = e + xe − x −

0

x

0

xu(t)dt

9. u(x) = 1 + sin x + x + x2 − x cos x − 10. u(x) = ex − xex + sin x + x cos x +

x

xu(t)dt

0

x

xu(t)dt

0

x 1 3 xu(t)dt x + cosh x + x sinh x − 2 0 x

esin t u(t)dt 12. u(x) = cos x − 1 − esin x x − x

11. u(x) = 1 + x + x2 +

0

13. u(x) = sec x2 − 1 − etan x x − x 14. u(x) = cosh x +

x

0

x x 1 − esinh x + 2 2

1 1 e − ecosh x + 10 10 x 1 tu(t)dt 16. u(x) = x3 − x5 + 5 10 0

15. u(x) = sinh x +

etan t u(t)dt x

0

esinh t u(t)dt x

0

ecosh t u(t)dt

78

3 Volterra Integral Equations

3.2.3 The Noise Terms Phenomenon It was shown before that the modiﬁed decomposition method presents a reliable tool for accelerating the computational work. However, a proper selection of f1 (x) and f2 (x) is essential for a successful use of this technique. A useful tool that will accelerate the convergence of the Adomian decomposition method is developed. The new technique depends mainly on the so-called noise terms phenomenon that demonstrates a fast convergence of the solution. The noise terms phenomenon can be used for all diﬀerential and integral equations. The noise terms, if existed between the components u0 (x) and u1 (x), will provide the exact solution by using only the ﬁrst two iterations. In what follows, we outline the main concepts of the noise terms : 1. The noise terms are deﬁned as the identical terms with opposite signs that arise in the components u0 (x) and u1 (x). Other noise terms may appear between other components. As stated above, these identical terms with opposite signs may exist for some equations, and may not appear for other equations. 2. By canceling the noise terms between u0 (x) and u1 (x), even though u1 (x) contains further terms, the remaining non-canceled terms of u0 (x) may give the exact solution of the integral equation. The appearance of the noise terms between u0 (x) and u1 (x) is not always suﬃcient to obtain the exact solution by canceling these noise terms. Therefore, it is necessary to show that the non-canceled terms of u0 (x) satisfy the given integral equation. On the other hand, if the non-canceled terms of u0 (x) did not satisfy the given integral equation, or the noise terms did not appear between u0 (x) and u1 (x), then it is necessary to determine more components of u(x) to determine the solution in a series form as presented before. 3.It was formally shown that the noise terms appear for speciﬁc cases of inhomogeneous diﬀerential and integral equations, whereas homogeneous equations do not give rise to noise terms. The conclusion about the selfcanceling noise terms was based on solving several speciﬁc diﬀerential and integral models. However, a proof for this conclusion was not given. For further readings about the noise terms phenomenon, see [7,10]. 4. It was formally proved in [7,10] that the appearance of the noise terms is governed by a necessary condition. The conclusion made in [7,10] is that the zeroth component u0 (x) must contain the exact solution u(x) among other terms. In addition, it was proved that the inhomogeneity condition of the equation does not always guarantee the appearance of the noise terms as examined in [10]. A useful summary about the noise terms phenomenon can be drawn as follows:

3.2 Volterra Integral Equations of the Second Kind

79

1. The noise terms are deﬁned as the identical terms with opposite signs that may appear in the components u0 (x) and u1 (x) and in the other components as well. 2. The noise terms appear only for speciﬁc types of inhomogeneous equations whereas noise terms do not appear for homogeneous equations. 3. Noise terms may appear if the exact solution of the equation is part of the zeroth component u0 (x). 4. Veriﬁcation that the remaining non-canceled terms satisfy the integral equation is necessary and essential. The phenomenon of the useful noise terms will be explained by the following illustrative examples. Example 3.11 Solve the Volterra integral equation by using noise terms phenomenon: 3 x tu(t)dt. (3.69) u(x) = 8x + x3 − 8 0 Following the standard Adomian method we set the recurrence relation: u0 (x) = 8x + x3 , 3 x (3.70) uk+1 (x) = − tuk (t)dt, k 0. 8 0 This gives u0 (x) = 8x + x3 , (3.71) 3 x 3 tu0 (t)dt = − x5 − x3 . u1 (x) = − 8 0 40 The noise terms ±x3 appear in u0 (x) and u1 (x). Canceling this term from the zeroth component u0 (x) gives the exact solution: u(x) = 8x, (3.72) that satisﬁes the integral equation. Notice that if the modiﬁed method is used, we select u0 (x) = 8x. As a result, we ﬁnd that u1 (x) = 0. This in turn gives the same result. Example 3.12 Solve the Volterra integral equation by using noise terms phenomenon: x 1 (x − t)2 u(t)dt. (3.73) u(x) = −2 + x + x2 + x4 + sin x + 2 cos x − 12 0 Following the standard Adomian method we set the recurrence relation: 1 u0 (x) = −2 + x + x2 + x4 + sin x + 2 cos x, 12 x (3.74) 2 uk+1 (x) = − (x − t) uk (t)dt, k 0. 0

This gives

80

3 Volterra Integral Equations

1 u0 (x) = −2 + x + x2 + x4 + sin x + 2 cos x, 12 3 x tu0 (t)dt u1 (x) = − 8 0 1 1 2 1 7 = 2 − x2 − x4 − 2 cos x + 4 sin x − x5 + x3 − x − 4x. 12 30 3 1260 (3.75) 1 4 x and ±2 cos x appear in u0 (x) and u1 (x). The noise terms ∓2, ±x2 , ± 12 Canceling these terms from the zeroth component u0 (x) gives the exact solution u(x) = x + sin x, (3.76) that satisﬁes the integral equation. It is to be noted that the other terms of u1 (x) vanish in the limit with other terms of the other components. Example 3.13 Solve the Volterra integral equation by using noise terms phenomenon x 1 1 2 u(t)dt. (3.77) u(x) = x − sinh(2x) + sinh x + 2 4 0 We set the recurrence relation: 1 1 u0 (x) = x − sinh(2x) + sinh2 x, 2 4 x (3.78) uk (t)dt, k 0. uk+1 (x) = 0

This gives 1 1 x − sinh(2x) + sinh2 x, 2 4 x 1 1 u1 (x) = u0 (t)dt = − x + sinh(2x) + 2 4 0 1 1 The noise terms ± 2 x and ∓ 4 sinh(2x) appear these terms from the zeroth component u0 (x) u0 (x) =

(3.79) 1 1 2 1 + x − cosh2 x. 4 4 4 in u0 (x) and u1 (x). Canceling gives the exact solution:

u(x) = sinh2 x, that satisﬁes the integral equation.

(3.80)

Example 3.14 Show that the exact solution for the Volterra integral equation: x 1 u(t)dt, u(x) = −1 + x + x2 + 2ex − 2 0 cannot be obtained by using the noise terms phenomenon. We set the recurrence relation: 1 u0 (x) = −1 + x + x2 + 2ex , 2 x uk (t)dt, k 0. uk+1 (x) = − 0

(3.81)

(3.82)

3.2 Volterra Integral Equations of the Second Kind

81

This gives 1 u0 (x) = −1 + x + x2 + 2ex , 2 x (3.83) 1 1 u1 (x) = − u0 (t)dt = − x2 − 2ex + 2 + x − x3 . 2 6 0 1 2 x The noise terms ± 2 x and ±2e appear in u0 (x) and u1 (x). Canceling these terms from the zeroth component u0 (x) gives u ˜(x) = x − 1, (3.84) that does not satisfy the integral equation. This conﬁrms our belief that the non-canceled terms of u0 (x) do not always give the exact solution, and therefore justiﬁcation is necessary. The exact solution is given by u(x) = x + ex ,

(3.85)

that can be easily obtained by using the modiﬁed decomposition method by setting f1 (x) = x + ex . Exercises 3.2.3 Use the noise terms phenomenon to solve the following Volterra integral equations: x x 1. u(x) = 6x + 3x2 − u(t)dt 2. u(x) = 6x + 3x2 − xu(t)dt 0

3. u(x) = 6x + 2x3 −

x

0

0

tu(t)dt 4. u(x) = x + x2 − 2x3 − x4 + 12

5. u(x) = −2 + x2 + sin x + 2 cos x − 6. u(x) = 2x − 2 sin x + cos x −

x

0

x 0

(x − t)2 u(t)dt

(x − t)2 u(t)dt

7. u(x) = sinh x + x sinh x − x2 cosh x +

x

xtu(t)dt

0

8. u(x) = x + cosh x + x2 sinh x − x cosh x − 9. u(x) = sec2 x − tan x +

x

0

xtu(t)dt

x

0

u(t)dt

1 1 10. u(x) = − x − sin(2x) + cos2 x + 2 4

x 0

u(t)dt

x 1 1 u(t)dt 11. u(x) = − x + sin(2x) + sin2 x + 2 4 0 x 12. u(x) = −x + tan x + tan2 x − u(t)dt 0

x

0

(x − t)u(t)dt

82

3 Volterra Integral Equations

3.2.4 The Variational Iteration Method In this section we will study the newly developed variational iteration method that proved to be eﬀective and reliable for analytic and numerical purposes. The variational iteration method (VIM) established by Ji-Huan He [11–12] is now used to handle a wide variety of linear and nonlinear, homogeneous and inhomogeneous equations. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the so called Adomian polynomials that we need for nonlinear problems. Moreover, the method gives the solution in a series form that converges to the closed form solution if an exact solution exists. The obtained series can be employed for numerical purposes if exact solution is not obtainable. In what follows, we present the main steps of the method. Consider the diﬀerential equation: Lu + N u = g(t), (3.86) where L and N are linear and nonlinear operators respectively, and g(t) is the source inhomogeneous term. The variational iteration method presents a correction functional for equation (3.86) in the form: x un+1 (x) = un (x) + λ(ξ) (Lun (ξ) + N u ˜n (ξ) − g(ξ)) dξ, (3.87) 0

where λ is a general Lagrange’s multiplier, noting that in this method λ may be a constant or a function, and u ˜n is a restricted value that means it behaves as a constant, hence δ˜ un = 0, where δ is the variational derivative. The Lagrange multiplier λ can be identiﬁed optimally via the variational theory as will be seen later. For a complete use of the variational iteration method, we should follow two steps, namely: 1. the determination of the Lagrange multiplier λ(ξ) that will be identiﬁed optimally, and 2. with λ determined, we substitute the result into (3.87) where the restrictions should be omitted. Taking the variation of (3.87) with respect to the independent variable un we ﬁnd x δun+1 δ =1+ λ(ξ) (Lun (ξ) + N u ˜n (ξ) − g(ξ)) dξ , (3.88) δun δun 0 or equivalently x λ(ξ) (Lun (ξ)) dξ . (3.89) δun+1 = δun + δ 0

3.2 Volterra Integral Equations of the Second Kind

83

Integration by parts is usually used for the determination of the Lagrange multiplier λ(ξ). In other words we can use x x λ(ξ)un (ξ)dξ = λ(ξ)un (ξ) − λ (ξ)un (ξ)dξ, 0

0

x

0 x 0

λ(ξ)un (ξ)dξ = λ(ξ)un (ξ) − λ (ξ)un (ξ) + λ(ξ)u n (ξ)dξ

0

x

x

0

λ (ξ)un (ξ)dξ,

= λ(ξ)un (ξ) − λ (ξ)un (ξ) + λ (ξ)un (ξ) −

x

(3.90)

λ (ξ)un (ξ)dξ,

0

λ(ξ)u(iv) n (ξ)dξ = λ(ξ)un (ξ) − λ (ξ)un (ξ) + λ (ξ)un (ξ) − λ un (ξ)

x

+ 0

λ(iv) (ξ)un (ξ)dξ,

and so on. These identities are obtained by integrating by parts. For example, if Lun (ξ) = un (ξ) in (3.89), then (3.89) becomes x δun+1 = δun + δ λ(ξ) (Lun (ξ)) dξ .

(3.91)

Integrating the integral of (3.91) by parts using (3.90) we obtain x λ (ξ)δun (ξ)dξ, δun+1 = δun + δλ(ξ)un (ξ) −

(3.92)

0

0

or equivalently

δun+1 = δun (ξ)(1 + λ|ξ=x ) −

x 0

λ δun dξ.

(3.93)

The extremum condition of un+1 requires that δun+1 = 0. This means that the left hand side of (3.93) is zero, and as a result the right hand side should be 0 as well. This yields the stationary conditions: 1 + λ|ξ=x = 0,

λ |ξ=x = 0.

(3.94)

This in turn gives λ = −1.

(3.95)

As a second example, if Lun (ξ) = un (ξ) in (3.89), then (3.89) becomes x λ(ξ) (Lun (ξ)) dξ . (3.96) δun+1 = δun + δ 0

Integrating the integral of (3.96) by parts using (3.90) we obtain x λ δun dξ, δun+1 = δun + δλ((un ) )x0 − (λ δun)x0 +

(3.97)

0

or equivalently δun+1 = δun (ξ)(1 − λ |ξ=x ) + δλ((un ) |ξ=x ) +

x 0

λ δun dξ,

(3.98)

84

3 Volterra Integral Equations

The extremum condition of un+1 requires that δun+1 = 0. This means that the left hand side of (3.98) is zero, and as a result the right hand side should be 0 as well. This yields the stationary conditions: 1 − λ |ξ=x = 0, λ|ξ=x = 0, λ |ξ=x = 0. (3.99) This in turn gives λ = ξ − x.

(3.100)

Having determined the Lagrange multiplier λ(ξ), the successive approximations un+1 , n 0, of the solution u(x) will be readily obtained upon using selective function u0 (x). However, for fast convergence, the function u0 (x) should be selected by using the initial conditions as follows: u0 (x) = u(0), for ﬁrst order un , u0 (x) = u(0) + xu (0), for second order un , (3.101) 1 u0 (x) = u(0) + xu (0) + x2 u (0), for third order u n, 2! .. . and so on. Consequently, the solution u(x) = lim un (x). n→∞

(3.102)

In other words, the correction functional (3.87) will give several approximations, and therefore the exact solution is obtained as the limit of the resulting successive approximations. The determination of the Lagrange multiplier plays a major role in the determination of the solution of the problem. In what follows, we summarize some iteration formulae that show ODE, its corresponding Lagrange multipliers, and its correction functional respectively:

⎧ ⎪ ⎨ u + f (u(ξ), u (ξ)) = 0, λ = −1, x (i) ⎪ [un + f (un , un )] dξ, ⎩ un+1 = un − 0

⎧ ⎪ ⎨ u + f (u(ξ), u (ξ), u (ξ)) = 0, λ = (ξ − x), x (ii) ⎪ (ξ − x) [u ⎩ un+1 = un + n + f (un , un , un )] dξ, 0

⎧ 1 2 ⎪ ⎪ ⎨ u + f (u(ξ), u (ξ), u (ξ), u (ξ)) = 0, λ = − 2! (ξ − x) , x (iii) 1 ⎪ ⎪ ⎩ un+1 = un − (ξ − x)2 [u n + f (un , . . . , un )] dξ, 2! 0 ⎧ 1 (iv) ⎪ ⎪ + f (u(ξ), u (ξ), u (ξ), u (ξ), u(iv) (ξ)) = 0, λ = (ξ − x)3 , ⎨u 3! x (iv) 1 ⎪ (iv) ⎪ ⎩ un+1 = un + (ξ − x)3 u n + f (un , un , . . . , un ) dξ, 0 3!

3.2 Volterra Integral Equations of the Second Kind

85

and generally ⎧ 1 ⎪ ⎪ (ξ − x)(n−1) , ⎨ u(n) + f (u(ξ), u (ξ), . . . , u(n) (ξ)) = 0, λ = (−1)n (n − 1)! (v) x 1 ⎪ (n) n ⎪ (ξ − x)(n−1) u ⎩ un+1 = un + (−1) n + f (un , . . . , un ) dξ, (n − 1)! 0 for n 1.

To use the variational iteration method for solving Volterra integral equations, it is necessary to convert the integral equation to an equivalent initial value problem or to an equivalent integro-diﬀerential equation. As deﬁned before, integro-diﬀerential equation is an equation that contains diﬀerential and integral operators in the same equation. The integro-diﬀerential equations will be studied in details in Chapter 5. The conversion process is presented in Section 2.5.1. However, for comparison reasons, we will examine the obtained initial value problem by two methods, namely, standard methods used for solving ODEs, and by using the variational iteration method as will be seen by the following examples. Example 3.15 Solve the Volterra integral equation by using the variational iteration method x u(t)dt. (3.103) u(x) = 1 + 0

Using Leibnitz rule to diﬀerentiate both sides of (3.103) gives u (x) − u(x) = 0.

(3.104)

Substituting x = 0 into (3.103) gives the initial condition u(0) = 1. Using the variational iteration method The correction functional for equation (3.104) is x un+1(x) = un (x) + λ(ξ)(un (ξ) − u ˜n (ξ))dξ.

(3.105)

Using the formula (i) given above leads to λ = −1.

(3.106)

0

Substituting this value of the Lagrange multiplier λ = −1 into the functional (3.105) gives the iteration formula: x (un (ξ) − un (ξ)) dξ. (3.107) un+1 (x) = un(x) − 0

As stated before, we can use the initial condition to select u0 (x) = u(0) = 1. Using this selection into (3.105) gives the following successive approximations: u0 (x) = 1, x (u0 (ξ) − u0 (ξ)) dξ = 1 + x, u1 (x) = 1 − 0

u2 (x) = 1 + x −

0

x

(u1 (ξ) − u1 (ξ)) dξ = 1 + x +

1 2 x , 2!

86

3 Volterra Integral Equations

u3 (x) = 1 + x +

1 2 x − 2!

x 0

(u2 (ξ) − u2 (ξ)) dξ = 1 + x +

1 2 1 x + x3 , 2! 3! (3.108)

and so on. The VIM admits the use of u(x) = limn→∞ un (x) 1 2 1 1 1 x + x3 + x4 + · · · + xn ), 2! 3! 4! n! that gives the exact solution by

(3.109)

= limn→∞ (1 + x +

u(x) = ex .

(3.110)

Using ODEs method The ODE (3.104) is of ﬁrst order, therefore the integrating factor μ(x) is given by x μ(x) = e (−1)dx = e−x . (3.111) For ﬁrst order ODE, we use the formula: x 1 μq(x)dx + C = Cex . (3.112) u(x) = μ To obtain the particular solution, we use the initial condition u(0) = 1 to ﬁnd that C = 1. This gives the particular solution: u(x) = ex .

(3.113)

Example 3.16 Solve the Volterra integral equation by using the variational iteration method x (x − t) u(t)dt. (3.114) u(x) = x + 0

Using Leibnitz rule to diﬀerentiate both sides of (3.114) once with respect to x gives the integro-diﬀerential equation: x u (x) = 1 + u(t)dt, u(0) = 0. (3.115) 0

However, by diﬀerentiating (3.115) with respect to x we obtain the diﬀerential equation: (3.116) u (x) = u(x). Substituting x = 0 into (3.114) and (3.115) gives the initial conditions u(0) = 0 and u (0) = 1. The resulting initial value problem, that consists of a second order ODE and initial conditions is given by (3.117) u (x) − u(x) = 0, u(0) = 0, u (0) = 1. The integro-diﬀerential equation (3.115) and the initial value problem (3.116) will be handled independently by using the variational iteration method. Using the variational iteration method (i) We ﬁrst start using the variational iteration method to handle the integrodiﬀerential equation (3.115) given by

3.2 Volterra Integral Equations of the Second Kind

u (x) = 1 +

87

x

u(t)dt, u(0) = 0.

(3.118)

0

The correction functional for Eq. (3.118) is x λ(ξ) un (ξ) − 1 − un+1 (x) = un (x) + 0

ξ 0

u ˜n (r)dr

dξ.

(3.119)

Using the formula (i) for λ we ﬁnd that λ = −1. (3.120) Substituting this value of the Lagrange multiplier λ = −1 into the functional (3.119) gives the iteration formula: x ξ un (ξ) − 1 − un+1 (x) = un (x) − un (r)dr dξ. (3.121) 0

0

We can use the initial conditions to select u0 (x) = u(0) = 0. Using this selection into (3.121) gives the following successive approximations: u0 (x) = 0, x ξ u1 (x) = x − u0 (r)dr dξ = x, u0 (ξ) − 1 − 0

u2 (x) = x −

0

u3 (x) = x −

x

0

x

0

u1 (ξ)

−1−

u2 (ξ)

−1−

u1 (r)dr

0

0

ξ

dξ = x +

ξ

u2 (r)dr

1 3 x , 3!

(3.122)

1 1 dξ = x + x3 + x5 , 3! 5!

.. . 1 3 1 1 1 x + x5 + x 7 + · · · + x2n+1 . 3! 5! 7! (2n + 1)! The VIM admits the use of u(x) = lim un (x), un (x) = x +

n→∞

(3.123)

that gives the exact solution by u(x) = sinh x. (3.124) (ii) We can obtain the same result by applying the variational iteration method to handle the initial value problem (3.117) given by (3.125) u (x) − u(x) = 0, u(0) = 0, u (0) = 1. The correction functional for Eq. (3.117) is x un+1 (x) = un (x) + λ(ξ) (un (ξ) − u ˜n (ξ)) dξ.

(3.126)

0

Using the formula (ii) given above leads to λ = ξ − x.

(3.127)

88

3 Volterra Integral Equations

Substituting this value of the Lagrange multiplier λ = ξ−x into the functional (3.126) gives the iteration formula: x un+1 (x) = un (x) + (ξ − x) (un (ξ) − un (ξ)) dξ. (3.128) 0

We can use the initial conditions to select u0 (x) = u(0) + xu (0) = x. Using this selection into (3.128) gives the following successive approximations u0 (x) = x, x 1 (ξ − x) (u0 (ξ) − u0 (ξ)) dξ = x + x3 , u1 (x) = x + 3! 0 x 1 1 1 (ξ − x) (u1 (ξ) − u1 (ξ)) dξ = x + x3 + x5 , u2 (x) = x + x3 + 3! 3! 5! 0 x 1 1 u3 (x) = x + x3 + x5 + (ξ − x) (u2 (ξ) − u2 (ξ)) dξ 3! 5! 0 1 5 1 7 1 3 = x+ x + x + x , 3! 5! 7! .. . 1 1 1 1 un (x) = x + x3 + x5 + x7 + · · · + x2n+1 . 3! 5! 7! (2n + 1)! (3.129) The VIM admits the use of u(x) = lim un (x), (3.130) n→∞

that gives the exact solution by u(x) = sinh x. (3.131) Standard methods for solving ODEs The initial value problem (3.125) is of second order, therefore the auxiliary equation is of the form (3.132) r2 − 1 = 0, that gives r = ±1. This in turn gives the general solution by u(x) = A sinh x + B cosh x.

(3.133)

To obtain the particular solution, we use the initial conditions u(0) = 0, u (0) = 1 to ﬁnd that the particular solution is given by u(x) = sinh x.

(3.134)

Example 3.17 Solve the Volterra integral equation by using the variational iteration method x 1 (x − t)u(t)dt. (3.135) u(x) = 1 + x + x3 − 3! 0 Using Leibnitz rule to diﬀerentiate both sides of (3.135) once with respect to x gives the integro-diﬀerential equation:

3.2 Volterra Integral Equations of the Second Kind x 1 2 u(t)dt, u(0) = 1, x − 2! 0 and by diﬀerentiating again we obtain the initial value problem

u (x) = 1 +

89

u (x) + u(x) = x, u(0) = 1, u (0) = 1.

(3.136)

(3.137)

Using the variational iteration method (i) We ﬁrst start using the variational iteration method to handle the integrodiﬀerential equation (3.136) given by x 1 u(t)dt, u(0) = 1, (3.138) u (x) = 1 + x2 − 2! 0 The correction functional for Eq. (3.138) is x ξ 1 2 λ(ξ) un (ξ) − 1 − ξ + u ˜n (r)dr dξ. (3.139) un+1 (x) = un (x) − 2 0 0 Using the formula (i) for λ we ﬁnd that λ = −1.

(3.140)

Substituting this value of the Lagrange multiplier λ = −1 into the functional (3.139) gives the iteration formula x ξ 1 2 un (r)dr dξ. (3.141) un (ξ) − 1 − ξ + un+1 (x) = un (x) − 2 0 0 We can use the initial conditions to select u0 (x) = u(0) = 1. Using this selection into (3.141) gives the following successive approximations u0 (x) = 1, x ξ 1 2 u0 (r)dr dξ u0 (ξ) − 1 − ξ + u1 (x) = x − 2 0 0 1 1 = 1 + x − x2 + x 3 , 2! 3! x ξ 1 u2 (x) = x − u1 (r)dr dξ u1 (ξ) − 1 − ξ 2 + 2 0 0 1 1 1 (3.142) = 1 + x − x2 + x 4 − x5 , 2! 4! 5! x ξ 1 u2 (ξ) − 1 − ξ 2 + u3 (x) = x − u2 (r)dr dξ 2 0 0 1 1 1 = 1 + x − x2 + x 4 − x6 , 2! 4! 6! .. . 1 1 1 (−1)n 2n x . un (x) = x + 1 − x2 + x4 − x6 + · · · + 2! 4! 6! (2n)! The VIM admits the use of

90

3 Volterra Integral Equations

u(x) = lim un (x), n→∞

(3.143)

that gives the exact solution by u(x) = x + cos x. (3.144) (ii) We next use the variational iteration method for solving the initial value problem u (x) + u(x) = x, u(0) = 1, u (0) = 1. (3.145) The correction functional for Eq. (3.145) is x λ(ξ) (un (ξ) + u ˜n (ξ) − ξ) dξ. un+1 (x) = un (x) +

(3.146)

0

Using the formula (ii) given above leads to λ = ξ − x.

(3.147)

Substituting this value of the Lagrange multiplier λ = ξ−x into the functional (3.146) gives the iteration formula x (ξ − x) (un (ξ) + un (ξ) − ξ) dξ. (3.148) un+1 (x) = un (x) + 0

We can use the initial conditions to select u0 (x) = u(0) + xu (0) = 1 + x. Using this selection into (3.148) gives the following successive approximations u0 (x) = 1 + x, x (ξ − x) (u0 (ξ) + u0 (ξ) − ξ) dξ u1 (x) = 1 + x + 0

1 = 1 + x − x2 , 2!

x 1 2 x + (ξ − x) (u1 (ξ) + u1 (ξ) − ξ) dξ 2! 0 1 4 1 2 = 1+x− x + x , (3.149) 2! 4! x 1 2 1 4 u3 (x) = 1 + x − x + x + (ξ − x) (u2 (ξ) + u2 (ξ) − ξ) dξ 2! 4! 0 1 1 1 = 1 + x − x2 + x4 − x6 2! 4! 6! .. . 1 1 1 1 (−1)n 2n . x un (x) = x + 1 − x2 + x4 − x6 + x8 − · · · + 2! 4! 6! 8! (2n)! Proceeding as before, the VIM gives the exact solution by u(x) = x + cos x. (3.150) u2 (x) = 1 + x −

Standard methods for solving ODEs The ODE (3.169) is of second order and nonhomogeneous. The auxiliary equation for the homogeneous part is of the form

3.2 Volterra Integral Equations of the Second Kind

91

r2 + 1 = 0,

(3.151)

2

that gives r = ±i, i = −1. The general solution is given by u(x) = uc + up , (3.152) u(x) = A cos x + B sin x + a + bx, where uc is the complementary solution, and up is a particular solution. Using ODE methods and initial conditions we ﬁnd that B = a = 0, and A = b = 1. The particular solution is given by u(x) = x + cos x.

(3.153)

Example 3.18 Solve the Volterra integral equation by using the variational iteration method 1 2 1 x u(x) = 1 + x + x + (x − t)2 u(t)dt. (3.154) 2 2 0 Using Leibnitz rule to diﬀerentiate both sides of (3.154) three times with respect to x gives the two integro-diﬀerential equations x (x − t)u(t)dt, u(0) = 1 u (x) = 1 + x + 0 (3.155) x u (x) = 1 + u(t)dt, u(0) = 1, u (0) = 1. 0

and the third order initial value problem u (x) = u(x), u(0) = u (0) = u (0) = 1. Using the variational iteration method VIM

(3.156)

(i) We ﬁrst note that we obtained two equivalent integro-diﬀerential equations (3.155). We will apply the VIM to these two equations. We ﬁrst start using the VIM to handle the integro-diﬀerential equation x (x − t)u(t)dt, u(0) = 1. (3.157) u (x) = 1 + x + 0

The correction functional for Eq. (3.157) is x ξ λ(ξ) un (ξ) − 1 − ξ − (ξ − r)˜ un (r)dr dξ. un+1 (x) = un (x) + 0

0

(3.158) Proceeding as before we ﬁnd λ = −1,

(3.159)

that gives the iteration formula x ξ un (ξ) − 1 − ξ − un+1 (x) = un (x) − (ξ − r)un (r)dr dξ. 0

(3.160)

0

We can use the initial conditions to select u0 (x) = u(0) = 1. Using this selection into (3.160) gives the following successive approximations

92

3 Volterra Integral Equations

u0 (x) = 1, u1 (x) = 1 −

x

0

= 1+x+ u2 (x) = 1 −

x

0

u0 (ξ)

−1−ξ−

1 1 2 x + x3 , 2! 3! u1 (ξ)

−1−ξ−

(ξ − r)u0 (r)dr dξ

0

0

ξ

ξ

(3.161)

(ξ − r)u1 (r)dr dξ

1 1 1 1 1 = 1 + x + x2 + x3 + x4 + x5 + x6 , 2! 3! 4! 5! 6! .. . 1 2 1 1 1 1 1 x + x3 + x4 + x5 + x6 + · · · + xn . 2! 3! 4! 5! 6! n! This in turn gives the exact solution by un (x) = 1 + x +

u(x) = ex .

(3.162)

(ii) We next consider the integro-diﬀerential equation x u(t)dt, u(0) = 1, u (0) = 1. u (x) = 1 +

(3.163)

0

The correction functional for Eq. (3.163) is x λ(ξ) un (ξ) − 1 − un+1 (x) = un (x) + 0

ξ 0

u ˜n (r)dr

dξ.

(3.164)

Notice that the integro-diﬀerential equation is of second order. Therefore, we can show that λ = ξ − x, (3.165) that gives the iteration formula x ξ (ξ − r)un (r)dr) dξ. (ξ − x)(un (ξ) − 1 − un+1 (x) = un (x) + 0

0

(3.166) We can use the initial conditions to select u0 (x) = 1 + x. Using this selection into (3.166) gives the following successive approximations u0 (x) = 1 + x, 1 2 1 1 x + x3 + x4 , 2! 3! 4! 1 1 1 u2 (x) = 1 + x + x2 + x3 + x4 + 2! 3! 4! .. . 1 1 1 un (x) = 1 + x + x2 + x3 + x4 + 2! 3! 4! This in turn gives the exact solution by u1 (x) = 1 + x +

1 5 1 1 x + x6 + x7 , 5! 6! 7! 1 5 1 1 x + x6 + · · · + xn . 5! 6! n!

(3.167)

3.2 Volterra Integral Equations of the Second Kind

u(x) = ex .

93

(3.168)

(iii) We next use the variational iteration method for solving the initial value problem u (x) − u(x) = 0, u(0) = u (0) = u (0) = 1. (3.169) x λ(ξ) (u ˜n (ξ)) dξ. (3.170) un+1 (x) = un (x) + n (ξ) − u 0

Using the formula (iii) given above for λ leads to 1 λ = − (ξ − x)2 . (3.171) 2! Substituting this value of the Lagrange multiplier into the functional (3.170) gives the iteration formula 1 x un+1 (x) = un (x) − (ξ − x)2 (u (3.172) n (ξ) − un (ξ)) dξ. 2! 0 As stated before, we can use the initial conditions to select 1 1 (3.173) u0 (x) = u(0) + xu (0) + u (0) = 1 + x + x2 . 2 2! Using this selection into (3.172) gives the following successive approximations 1 u0 (x) = 1 + x + x2 , 2! 1 1 1 1 u1 (x) = 1 + x + x2 + x3 + x4 + x5 , 2! 3! 4! 5! 1 2 1 3 1 4 1 1 1 1 u2 (x) = 1 + x + x + x + x + x5 + x6 + x7 + x8 , (3.174) 2! 3! 4! 5! 6! 7! 8! .. . 1 1 1 1 1 1 1 un (x) = 1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x8 + · · · 2! 3! 4! 5! 6! 7! 8! The VIM admits the use of u(x) = lim un (x), n→∞

(3.175)

that gives the exact solution by (3.176) u(x) = ex . Using standard methods for solving ODEs The ODE (3.169) is of third order, therefore the auxiliary equation is of the form r3 − 1 = 0, (3.177) that gives r = 1, − 21 ±

√

3 i, i2 2

= −1. The general solution is given by √ √ 3 3 x − 12 x x + C sin x . (3.178) B cos u(x) = Ae + e 2 2

To obtain the particular solution, we use the initial conditions to ﬁnd that the particular solution

94

3 Volterra Integral Equations

u(x) = ex .

(3.179)

It is interesting to point out that we need to use diﬀerent approaches to solve ODEs by standard methods, whereas the variational iteration method attacks all problems directly and in a straightforward manner.

Exercises 3.2.4 Use the variational iteration method to solve the following Volterra integral equations by converting the equation to initial value problem or to an equivalent integrodiﬀerential equation: x x 1 1 1. u(x) = 1 − u(t)dt 2. u(x) = x + x4 + x2 + x5 − u(t)dt 2 5 0 0 x x 1 1 u(t)dt 4. u(x) = 1 − x − x2 + (x − t)u(t)dt 3. u(x) = 1 − x2 + 2 2 0 0 x x 5. u(x) = 1 − (x − t)u(t)dt 6. u(x) = x + (x − t)u(t)dt 0

0

7. u(x) = 1 + 2x + 4

x

0

(x − t)u(t)dt 8. u(x) = 5 + 2x2 −

1 2 1 x x + (x − t)2 u(t)dt 2 2 0 1 x 1 10. u(x) = 1 + x + (x − t + 1)u(t)dt 2 2 0 x (x − t + 1)2 u(t)dt 11. u(x) = 1 + x2 −

x

0

(x − t)u(t)dt

9. u(x) = 1 + x +

0

1 1 12. u(x) = 1 − x2 + 2 6

x

0

(x − t)3 u(t)dt

13. u(x) = 2 + x − 2 cos x − 14. u(x) = 1 − 2 sinh x + 15. u(x) = 1 +

x

0

x 0

(x − t + 2)u(t)dt

(x − t + 2)u(t)dt

1 2 3 1 x+ x + 5 10 10

x

(x − t + 2)2 u(t)dt

0

3 2 1 3 1 x + x − 2 6 2 x 17. u(x) = 1 − x sin x + tu(t)dt

16. u(x) = 2 + 3x +

0

19. u(x) = −1 + ex +

1 2 x 1 x e − 2 2

20. u(x) = 1 − x sin x + x cos x +

0

(x − t + 2)2 u(t)dt

18. u(x) = x cosh x −

x

0 x 0

x

tu(t)dt tu(t)dt

x 0

tu(t)dt

3.2 Volterra Integral Equations of the Second Kind

95

3.2.5 The Successive Approximations Method The successive approximations method, also called the Picard iteration method provides a scheme that can be used for solving initial value problems or integral equations. This method solves any problem by ﬁnding successive approximations to the solution by starting with an initial guess, called the zeroth approximation. As will be seen, the zeroth approximation is any selective real-valued function that will be used in a recurrence relation to determine the other approximations. Given the linear Volterra integral equation of the second kind x u(x) = f (x) + λ K(x, t)u(t)dt, (3.180) 0

where u(x) is the unknown function to be determined, K(x, t) is the kernel, and λ is a parameter. The successive approximations method introduces the recurrence relation x

un (x) = f (x) + λ

0

K(x, t)un−1 (t)dt, n 1,

(3.181)

where the zeroth approximation u0 (x) can be any selective real valued function. We always start with an initial guess for u0 (x), mostly we select 0, 1, x for u0 (x), and by using (3.181), several successive approximations uk , k 1 will be determined as x

u1 (x) = f (x) + λ

u2 (x) = f (x) + λ

.. . un (x) = f (x) + λ

x

K(x, t)u1 (t)dt,

0

u3 (x) = f (x) + λ

K(x, t)u0 (t)dt,

0

x

K(x, t)u2 (t)dt,

0

0

(3.182)

x

K(x, t)un−1 (t)dt.

The question of convergence of un (x) is justiﬁed by noting the following theorem. Theorem 3.1 If f (x) in (3.181) is continuous for the interval 0 x a, and the kernel K(x, t) is also continuous in the triangle 0 x a, 0 t x, the sequence of successive approximations un (x), n 0 converges to the solution u(x) of the integral equation under discussion. It is interesting to point out that the variational iteration method admits the use of the iteration formula: x ∂un (ξ) −u ˜n (ξ) dξ. λ(ξ) (3.183) un+1 (x) = un (x) + ∂ξ 0 whereas the successive approximations method uses the iteration formula

96

3 Volterra Integral Equations

un (x) = f (x) + λ

x 0

K(x, t)un−1 (t)dt, n 1.

(3.184)

The diﬀerence between the two formulas can be summarized as follows: 1. The ﬁrst formula contains the Lagrange multiplier λ that should be determined ﬁrst before applying the formula. The successive approximations formula does not require the use of λ. 2. The ﬁrst variational iteration formula allows the use of the restriction u ˜n (ξ) where δ˜ un (ξ) = 0. The second formula does require this restriction. 3. The ﬁrst formula is applied to an equivalent ODE of the integral equation, whereas the second formula is applied directly to the iteration formula of the integral equation itself. The successive approximations method, or the Picard iteration method will be illustrated by the following examples. Example 3.19 Solve the Volterra integral equation by using the successive approximations method x u(x) = 1 − (x − t)u(t)dt. (3.185) 0

For the zeroth approximation u0 (x), we can select (3.186) u0 (x) = 1. The method of successive approximations admits the use of the iteration formula x (x − t)un (t)dt, n 0. (3.187) un+1 (x) = 1 − 0

Substituting (3.186) into (3.187) we obtain x 1 (x − t)u0 (t)dt = 1 − x2 , u1 (x) = 1 − 2! 0 x 1 u2 (x) = 1 − (x − t)u1 (t)dt = 1 − x2 + 2! 0 x 1 (x − t)u2 (t)dt = 1 − x2 + u3 (x) = 1 − 2! 0 x 1 (x − t)u3 (t)dt = 1 − x2 + u4 (x) = 1 − 2! 0 .. .

1 4 x , 4! 1 4 1 x − x6 , 4! 6!

(3.188)

1 4 1 1 x − x6 + x8 , 4! 6! 8!

Consequently, we obtain un+1 (x) =

n

(−1)k

k=0

x2k . (2k)!

(3.189)

The solution u(x) of (3.185) u(x) = lim un+1 (x) = cos x. n→∞

(3.190)

3.2 Volterra Integral Equations of the Second Kind

97

Example 3.20 Solve the Volterra integral equation by using the successive approximations method 1 1 x u(x) = 1 + x + x2 + (x − t)2 u(t)dt. (3.191) 2 2 0 For the zeroth approximation u0 (x), we can select u0 (x) = 0.

(3.192)

The method of successive approximations admits the use of the iteration formula 1 1 x (x − t)2 un(t)dt, n 0. (3.193) un+1 (x) = 1 + x + x2 + 2 2 0 Substituting (3.192) into (3.193) we obtain 1 u1 (x) = 1 + x + x2 , 2! 1 1 1 1 (3.194) u2 (x) = 1 + x + x2 + x3 + x4 + x5 , 2! 3! 4! 5! 1 1 1 1 1 1 1 u3 (x) = 1 + x + x2 + x3 + x4 + x5 + x6 + x7 + x8 , 2! 3! 4! 5! 6! 7! 8! .. . and so on. The solution u(x) of (3.191) is given by (3.195) u(x) = lim un+1 (x) = ex . n→∞

Example 3.21 Solve the Volterra integral equation by using the successive approximations method 1 1 x u(x) = −1 + ex + x2 ex − tu(t)dt. (3.196) 2 2 0 For the zeroth approximation u0 (x), we select u0 (x) = 0. We next use the iteration formula 1 1 x tun (t)dt, n 0. un+1 (x) = −1 + ex + x2 ex − 2 2 0 Substituting (3.197) into (3.198) we obtain 1 u1 (x) = −1 + ex + x2 ex , 2! 1 5 1 u2 (x) = −3 + x2 + ex (3 − 2x + x2 − x3 ), 4 4 4

(3.197) (3.198)

98

3 Volterra Integral Equations

1 2 1 3 1 4 1 5 u3 (x) = x 1 + x + x + x + x + x + · · · , (3.199) 2! 3! 4! 5! .. . 1 2 1 3 1 4 1 5 1 7 un+1 (x) = x 1 + x + x + x + x + x + x + · · · . 2! 3! 4! 5! 7! x Notice that we used the Taylor expansion for e to determine u3 (x), u4 (x), . . . The solution u(x) of (3.196) u(x) = lim un+1 (x) = xex . n→∞

(3.200)

Example 3.22 Solve the Volterra integral equation by using the successive approximations method x tu(t)dt. (3.201) u(x) = 1 − x sin x + x cos x + 0

For the zeroth approximation u0 (x), we may select u0 (x) = x. We next use the iteration formula un+1 (x) = 1 − x sin x + x cos x +

0

(3.202) x

tun (t)dt, n 0.

(3.203)

Substituting (3.202) into (3.203) gives 1 u1 (x) = 1 + x3 − x sin x + x cos x, 3 1 1 u2 (x) = 3 + x2 + x3 − (2 + 3x − x2 ) sin x − (2 − 3x − x2 ) cos x, 2 15 1 2 1 3 1 5 1 7 1 4 1 6 u3 (x) = x − x + x − x + 1 − x + x − x , 3! 5! 7! 2! 4! 6! .. . n n x2k+1 x2k + . un+1 (x) = (−1)k (−1)k (2k + 1)! (2k)! k=0 k=0 (3.204) Notice that we used the Taylor expansion for sin x and cos x to determine the approximations u3 (x), u4 (x), . . .. The solution u(x) of (3.201) is given by u(x) = lim un+1 (x) = sin x + cos x. n→∞

(3.205)

Exercises 3.2.5 Use the successive approximations method to solve the following Volterra integral equations: x x 1. u(x) = x + u(t)dt 2. u(x) = x + (x − t)u(t)dt 0

0

3.2 Volterra Integral Equations of the Second Kind 99 x x 1 3. u(x) = x3 − (x − t)u(t)dt 4. u(x) = 1 + 2x + 4 (x − t)u(t)dt 6 0 0 x x 1 5. u(x) = x3 + (x − t)u(t)dt 6. u(x) = 1 + x2 − (x − t + 1)2 u(t)dt 6 0 0 x 1 1 1 x 7. u(x) = x2 − (x − t)u(t)dt 8. u(x) = 1 − x2 + (x − t)3 u(t)dt 2 2 6 0 0 x x u(t)dt 10. u(x) = 1 − 2 sinh x + (x − t + 2)u(t)dt 9. u(x) = 1 + 3 0

2

11. u(x) = 3 + x −

x

0

13. u(x) = x cosh x − 15. u(x) = 1 −

x

0

(x − t)u(t)dt 12. u(x) = 1 − x sin x +

x

tu(t)dt

0

3t2 u(t)dt

14. u(x) = 1 − x −

18. u(x) = 1 + sinh x + sin x − cos x + cosh x −

x

0

20. u(x) = −x + 2 sinh x +

x 0

tu(t)dt

0

(x − t)u(t)dt

16. u(x) = 2x cosh x − 4

17. u(x) = 1 + sinh x − sin x + cos x − cosh x +

19. u(x) = 2 − 2 cos x −

0 x

x

0

tu(t)dt

x

0 x 0

u(t)dt u(t)dt

(x − t)u(t)dt

x 0

(x − t)u(t)dt

3.2.6 The Laplace Transform Method The Laplace transform method is a powerful technique that can be used for solving initial value problems and integral equations as well. We assume that the reader has used the Laplace transform method, and the inverse Laplace transform, for solving ordinary diﬀerential equations. The details and properties of the Laplace method can be found in ordinary diﬀerential equations texts. Before we start applying this method, we summarize some of the concepts presented in Section 1.5. In the convolution theorem for the Laplace transform, it was stated that if the kernel K(x, t) of the integral equation: x u(x) = f (x) + λ K(x, t)u(t)dt, (3.206) 0

depends on the diﬀerence x − t, then it is called a diﬀerence kernel. Examples of the diﬀerence kernel are ex−t , cos(x − t), and x − t. The integral equation can thus be expressed as x K(x − t)u(t)dt. (3.207) u(x) = f (x) + λ 0

100

3 Volterra Integral Equations

Consider two functions f1 (x) and f2 (x) that possess the conditions needed for the existence of Laplace transform for each. Let the Laplace transforms for the functions f1 (x) and f2 (x) be given by L{f1 (x)} = F1 (s), (3.208) L{f2 (x)} = F2 (s). The Laplace convolution product of these two functions is deﬁned by x f1 (x − t)f2 (t)dt, (3.209) (f1 ∗ f2 )(x) = 0

or (f2 ∗ f1 )(x) =

0

x

f2 (x − t)f1 (t)dt.

(3.210)

Recall that (f1 ∗ f2 )(x) = (f2 ∗ f1 )(x).

(3.211)

We can easily show that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x L{(f1 ∗ f2 )(x)} = L f1 (x − t)f2 (t)dt = F1 (s)F2 (s). (3.212) 0

Based on this summary, we will examine speciﬁc Volterra integral equations where the kernel is a diﬀerence kernel. Recall that we will apply the Laplace transform method and the inverse of the Laplace transform using Table 1.1 in Section 1.5. By taking Laplace transform of both sides of (3.207) we ﬁnd U (s) = F (s) + λK(s)U (s),

(3.213)

where U (s) = L{u(x)}, K(s) = L{K(x)}, F (s) = L{f (x)}. (3.214) Solving (3.213) for U (s) gives F (s) , λK(s) = 1. (3.215) U (s) = 1 − λK(s) The solution u(x) is obtained by taking the inverse Laplace transform of both sides of (3.215) where we ﬁnd F (s) −1 u(x) = L . (3.216) 1 − λK(s) Recall that the right side of (3.216) can be evaluated by using Table 1.1 in Section 1.5. The Laplace transform method for solving Volterra integral equations will be illustrated by studying the following examples. Example 3.23 Solve the Volterra integral equation by using the Laplace transform method x u(t)dt. (3.217) u(x) = 1 + 0

3.2 Volterra Integral Equations of the Second Kind

101

Notice that the kernel K(x − t) = 1, λ = 1. Taking Laplace transform of both sides (3.217) gives L{u(x)} = L{1} + L{1 ∗ u(x)}, (3.218) so that U (s) =

1 1 + U (s), s s

(3.219)

1 . s−1

(3.220)

or equivalently U (s) =

By taking the inverse Laplace transform of both sides of (3.220), the exact solution is therefore given by (3.221) u(x) = ex . Example 3.24 Solve the Volterra integral equation by using the Laplace transform method x (x − t)u(t)dt. (3.222) u(x) = 1 − 0

Notice that the kernel K(x − t) = (x − t), λ = −1. Taking Laplace transform of both sides (3.222) gives L{u(x)} = L{1} − L{(x − t) ∗ u(x)}, so that U (s) =

(3.223)

1 1 − 2 U (s), s s

(3.224)

s . s2 + 1

(3.225)

or equivalently U (s) =

By taking the inverse Laplace transform of both sides of (3.225), the exact solution u(x) = cos x, (3.226) is readily obtained. Example 3.25 Solve the Volterra integral equation by using the Laplace transform x 1 (x − t)u(t)dt. u(x) = x3 − 3! 0 Taking Laplace transform of both sides (3.227) gives 1 L{u(x)} = L{x3 } − L{(x − t) ∗ u(x)}. 3! This gives 3! 1 1 × 4 − 2 U (s), U (s) = 3! s s so that

method (3.227)

(3.228)

(3.229)

102

3 Volterra Integral Equations

U (s) =

1 1 1 = 2 − 2 . s2 (s2 + 1) s s +1

(3.230)

Taking the inverse Laplace transform of both sides of (3.230) gives the exact solution u(x) = x − sin x. (3.231) Example 3.26 Solve the Volterra integral equation by using the Laplace transform method x sin(x − t)u(t)dt. (3.232) u(x) = sin x + cos x + 2 0

Recall that we should use the linear property of the Laplace transforms. Taking Laplace transform of both sides (3.232) gives L{u(x)} = L{sin x + cos x} + 2L{sin(x − t) ∗ u(x)}, (3.233) so that U (s) =

s2

s 2 1 + 2 + 2 U (s), +1 s +1 s +1

(3.234)

or equivalently U (s) =

1 . s−1

(3.235)

Taking the inverse Laplace transform of both sides of (3.235) gives the exact solution (3.236) u(x) = ex .

Exercises 3.2.6 Use the Laplace transform method to solve the Volterra integral equations: x x 1. u(x) = x + (x − t)u(t)dt 2. u(x) = 1 − x − (x − t)u(t)dt 0

1 x 1 (x − t)3 u(t)dt 3. u(x) = 1 − x2 + 2 6 0 x 5. u(x) = x − 1 + (x − t)u(t)dt 0

6. u(x) = cos x − sin x + 2 7. u(x) = ex − cos x − 2

x

0 x

0

0

4. u(x) = 1 + 3

x 0

(x − t)u(t)dt

cos(x − t)u(t)dt

ex−t u(t)dt

9. u(x) = sin x + sinh x + cosh x − 2

x

cos (x − t)u(t)dt

0

10. u(x) = sinh x + cosh x − cos x − 2 11. u(x) = sin x − cos x + cosh x − 2

8. u(x) = 1 −

x

0 x

0

cos (x − t)u(t)dt

cosh (x − t)u(t)dt

x

0

(x − t)2 − 1 u(t)dt

3.2 Volterra Integral Equations of the Second Kind x 12. u(x) = sin x + cos x + sinh x − 2 cosh (x − t)u(t)dt

13. u(x) = 2ex − 2 − x + 15. u(x) = 2 − 2 cos x −

0

x

0 x

0

103

(x − t)u(t)dt 14. u(x) = 2 cosh x − 2 +

(x − t)u(t)dt

16. u(x) = 1 +

x 0

x

0

(x − t)u(t)dt

sin(x − t)u(t)dt

3.2.7 The Series Solution Method A real function u(x) is called analytic if it has derivatives of all orders such that the Taylor series at any point b in its domain n f (k) (b) (x − b)k , (3.237) u(x) = k! k=0

converges to f (x) in a neighborhood of b. For simplicity, the generic form of Taylor series at x = 0 can be written as ∞ an xn . (3.238) u(x) = n=0

In this section we will present a useful method, that stems mainly from the Taylor series for analytic functions, for solving Volterra integral equations. We will assume that the solution u(x) of the Volterra integral equation x K(x, t)u(t)dt, (3.239) u(x) = f (x) + λ 0

is analytic, and therefore possesses a Taylor series of the form given in (3.238), where the coeﬃcients an will be determined recurrently. Substituting (3.238) into both sides of (3.239) gives ∞ x ∞ n n an x = T (f (x)) + λ K(x, t) an t dt, (3.240) 0

n=0

or for simplicity we use 2

a0 + a1 x + a2 x + · · · = T (f (x)) + λ

n=0

0

x

K(x, t) a0 + a1 t + a2 t2 + · · · dt,

(3.241) where T (f (x)) is the Taylor series for f (x). The integral equation (3.239) will be converted to a traditional integral in (3.240) or (3.241) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integral in (3.240) or (3.241), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of

104

3 Volterra Integral Equations

like powers of x in both sides of the resulting equation to obtain a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (3.238). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. Example 3.27 Solve the Volterra integral equation by using the series solution method x u(x) = 1 + u(t)dt. (3.242) 0

Substituting u(x) by the series u(x) =

∞

an xn ,

(3.243)

n=0

into both sides of Eq. (3.242) leads to ∞ an xn = 1 + n=0

x

0

∞

an t

n

dt.

Evaluating the integral at the right side gives ∞ ∞ 1 an xn+1 , an xn = 1 + n + 1 n=0 n=0 that can be rewritten as a0 +

∞ n=1

a n xn = 1 +

(3.244)

n=0

∞ 1 an−1 xn , n n=1

(3.245)

(3.246)

or equivalently 1 1 a0 + a1 x + a2 x2 + a3 x3 + · · · = 1 + a0 x + a1 x2 + a2 x2 + · · · . (3.247) 2 3 In (3.245), the powers of x of both sides are diﬀerent, therefore, we make them the same by changing the index of the second sum to obtain (3.246). Equating the coeﬃcients of like powers of x in both sides of (3.246) gives the recurrence relation a0 = 1, (3.248) 1 an = an−1 , n 1. n where this result gives 1 (3.249) an = , n 0. n! Substituting this result into (3.243) gives the series solution:

3.2 Volterra Integral Equations of the Second Kind

105

∞ 1 n x , n! n=0

(3.250)

u(x) =

that converges to the exact solution u(x) = ex . It is interesting to point out that this result can be obtained by equating coeﬃcients of like terms in both sides of (3.247), where we ﬁnd a0 = 1, a1 = a0 = 1, 1 1 a 2 = a1 = , 2 2! (3.251) .. . 1 1 an−1 = . n n! This leads to the same result obtained before by solving the recurrence relation. an =

Example 3.28 Solve the Volterra integral equation by using the series solution method x (x − t)u(t)dt. (3.252) u(x) = x + 0

Substituting u(x) by the series u(x) =

∞

an xn ,

(3.253)

n=0

into both sides of Eq. (3.252) leads to x ∞ ∞ ∞ an xn = x + xan tn − an tn+1 dt. 0

n=0

n=0

Evaluating the right side leads to ∞ ∞ an xn = x +

1 an xn+2 , (n + 2)(n + 1) n=0

n=0

that can be rewritten as a0 + a1 x +

∞ n=2

an xn = x +

(3.254)

n=0

∞

1 an−2 xn , n(n − 1) n=2

(3.255)

(3.256)

or equivalently 1 1 1 a0 + a1 x + a2 x2 + a3 x3 + · · · = x + a0 x2 + a1 x3 + a2 x4 + · · · (3.257) 2 6 12 Equating the coeﬃcients of like powers of x in both sides of (3.256) gives the recurrence relation a0 = 0, a1 = 1, .. . (3.258) 1 an−2 , n 2. an = n(n − 1)

106

3 Volterra Integral Equations

This result can be combined to obtain 1 , n 0. an = (2n + 1)! Substituting this result into (3.253) gives the series solution ∞ 1 u(x) = x2n+1 , (2n + 1)! n=0 that converges to the exact solution u(x) = sinh x.

(3.259)

(3.260)

(3.261)

It is interesting to point out that this result can also be obtained by equating coeﬃcients of like terms in both sides of (3.257), where we ﬁnd 1 a0 = 0, a1 = 1, a2 = a0 = 0, 2 (3.262) 1 1 1 a2 = 0. a3 = a1 = , a4 = 6 3! 12 This leads to the same result obtained before by solving the recurrence relation. Example 3.29 Solve the Volterra integral equation by using the series solution method x tu(t)dt. (3.263) u(x) = 1 − x sin x + 0

For simplicity reasons, we will use few terms of the Taylor series for sin x and for the solution u(x) in (3.263) to ﬁnd a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · x (3.264) x3 =1−x x− t(a0 + a1 t + a2 t2 + · · · )dt. + ··· + 3! 0 Integrating the right side and collecting the like terms of x we ﬁnd 1 1 1 1 a0 +a1 x+a2 x2 +a3 x3 +a4 x4 +· · · = 1+( a0 −1)x2 + a1 x3 +( + a2 )x4 +· · · . 2 3 6 4 (3.265) Equating the coeﬃcients of like powers of x in (3.265) yields a1 = 0, a0 = 1, 1 1 1 a2 = a0 − 1 = − , a3 = a1 = 0, 2 2! 3 (3.266) 1 1 1 a 4 = + a2 = , 6 4 4! .. . and generally a2n+1 = 0,

a2n =

(−1)n , (2n)!

n 0.

(3.267)

3.2 Volterra Integral Equations of the Second Kind

107

The solution in a series form is given by 1 1 1 u(x) = 1 − x2 + x4 − x6 + · · · 2! 4! 6! that gives the exact solution by

(3.268)

u(x) = cos x.

(3.269)

Example 3.30 Solve the Volterra integral equation by using the series solution method x u(x) = 2ex − 2 − x + (x − t)u(t)dt. (3.270) 0

Proceeding as before, we will use few terms of the Taylor series for ex and for the solution u(x) in (3.270) to ﬁnd a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · x 1 1 (x − t)(a0 + a1 t + a2 t2 + · · · )dt. = x + x2 + x3 + x4 + · · · + 3 12 0 (3.271) Integrating the right side and collecting the like terms of x we ﬁnd a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · 1 1 1 1 1 + a1 x3 + + a2 x4 + · · · . = x + 1 + a 0 x2 + 2 3 6 12 12 (3.272) Equating the coeﬃcients of like powers of x in (3.272) yields 1 1 a0 = 0, a1 = 1, a2 = 1, a3 = , a4 = , . . . (3.273) 2! 3! and generally 1 an = , n 1, a0 = 0. (3.274) n! The solution in a series form is given by 1 1 1 (3.275) u(x) = x(1 + x + x2 + x3 + x4 + · · · ), 2! 3! 4! that converges to the exact solution u(x) = xex .

(3.276)

Exercises 3.2.7 Use the series solution method to solve the Volterra integral equations: x x 1. u(x) = 1 − u(t)dt 2. u(x) = 1 − (x − t)u(t)dt 0

3. u(x) = x +

0

x

0

(x − t)u(t)dt

5. u(x) = 1 + xex −

x

0

tu(t)dt

1 1 4. u(x) = 1 + x + 2 2 6. u(x) = 1 + 2x + 4

x

0 x 0

(x − t + 1)u(t)dt

(x − t)u(t)dt

108 7. u(x) = 3 + x2 −

x

0 x

9. u(x) = x cos x +

0

(x − t)u(t)dt

12. u(x) = 1 − x −

x 0

10. u(x) = x cosh x −

tu(t)dt

11. u(x) = 2 cosh x − 2 +

3 Volterra Integral Equations x 8. u(x) = 1 + 2 sin x − u(t)dt

x

0

0 x

0

tu(t)dt

(x − t)u(t)dt

(x − t)u(t)dt

13. u(x) = x − x ln(1 + x) +

x 1 3 2xu(t)dt x + x3 ln(1 + x) − 2 0 x 15. u(x) = sec x + tan x − sec tu(t)dt 16. u(x) = x +

14. u(x) = x2 −

0

x

0

u(t)dt

x

0

tan tu(t)dt

3.3 Volterra Integral Equations of the First Kind The standard form of the Volterra integral equations of the ﬁrst kind is given by x

K(x, t)u(t)dt,

f (x) =

(3.277)

0

where the kernel K(x, t) and the function f (x) are given real-valued functions, and u(x) is the function to be determined. Recall that the unknown function u(x) appears inside and outside the integral sign for the Volterra integral equations of the second kind, whereas it occurs only inside the integral sign for the Volterra integral equations of the ﬁrst kind. This equation of the ﬁrst kind motivated mathematicians to develop reliable methods for solving it. In this section we will discuss three main methods that are commonly used for handling the Volterra integral equations of the ﬁrst kind. Other methods are available in the literature but will not be presented in this text.

3.3.1 The Series Solution Method As in the previous section, we will consider the solution u(x) to be analytic, where it has derivatives of all orders, and it possesses Taylor series at x = 0 of the form ∞ u(x) = a n xn , (3.278) n=0

where the coeﬃcients an will be determined recurrently. Substituting (3.278) into (3.277) gives

3.3 Volterra Integral Equations of the First Kind

T (f (x)) =

x

K(x, t) 0

∞

109

an tn dt,

(3.279)

n=0

or for simplicity we can use x

K(x, t) a0 + a1 t + a2 t2 + · · · dt, T (f (x)) =

(3.280)

0

where T (f (x)) is the Taylor series for f (x). The integral equation (3.277) will be converted to a traditional integral in (3.279) or (3.280) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. The method is identical to that presented before for the Volterra integral equations of the second kind. We ﬁrst integrate the right side of the integral in (3.279) or (3.280), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x in both sides of the resulting equation to obtain a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (3.278). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. This method will be illustrated by discussing the following examples. Example 3.31 Solve the Volterra integral equation by using the series solution method x tu(t)dt. (3.281) sin x − x cos x = 0

Proceeding as before, only few terms of the Taylor series for sin x − x cos x and for the solution u(x) in (3.281) will be used. Integrating the right side we obtain x 1 3 1 5 1 7 x − x + x + ··· = t(a0 + a1 t + a2 t2 + a3 t3 + · · · )dt, 3 30 840 0 1 1 1 1 = a0 x2 + a1 x3 + a2 x4 + a3 x5 2 3 4 5 1 1 + a4 x6 + a5 x7 + · · · 6 7 (3.282) Equating the coeﬃcients of like powers of x in (3.282) yields 1 1 (3.283) a0 = 0, a1 = 1, a2 = 0, a3 = − , a4 = 0, a5 = · · · 3! 5! The solution in a series form is given by

110

3 Volterra Integral Equations

1 3 1 x + x5 + · · · 3! 5! that converges to the exact solution u(x) = sin x. u(x) = x −

(3.284) (3.285)

Example 3.32 Solve the Volterra integral equation by using the series solution method x (x − t)u(t)dt. (3.286) 2 + x − 2ex + xex = 0

Using few terms of the Taylor series for 2 + x − 2ex + xex and for the solution u(x) in (3.286), and by integrating the right side we ﬁnd x 1 1 1 3 x + x4 + x5 + · · · = (x − t)(a0 + a1 t + a2 t2 + · · · )dt 6 12 40 0 1 1 1 1 = a0 x2 + a1 x3 + a2 x4 + a3 x5 (3.287) 2 6 12 20 1 6 + a4 x + · · · 30 Equating the coeﬃcients of like powers of x in (3.287) yields 1 1 a0 = 0, a1 = 1, a2 = 1, a3 = , a4 = , · · · (3.288) 2! 3! The solution in a series form is given by 1 1 u(x) = x(1 + x + x2 + x3 + · · · ), (3.289) 2! 3! that converges to the exact solution (3.290) u(x) = xex . Example 3.33 Solve the Volterra integral equation by using the series solution method x 1 2 2 2tu(t)dt. (3.291) x − x − ln(1 + x) + x ln(1 + x) = 2 0 Using the Taylor series for x − 12 x2 − ln(1 + x) + x2 ln(1 + x) and proceeding as before we ﬁnd 2 1 2 3 1 4 x − x + x5 − x6 + · · · 3 4 15 12 x (3.292) = 2t(a0 + a1 t + a2 t2 + a3 t3 + · · · )dt 0

2 1 2 1 = a0 x2 + a1 x3 + a2 x4 + a3 x5 + a4 x6 + · · · 3 2 5 3 Equating the coeﬃcients of like powers of x in (3.292) yields 1 1 1 a0 = 0, a1 = 1, a2 = − , a3 = , a4 = − , . . . 2 3 4 The solution in a series form is given by

(3.293)

3.3 Volterra Integral Equations of the First Kind

111

1 1 1 u(x) = x − x2 + x3 − x4 + · · · 2 3 4 that converges to the exact solution u(x) = ln(1 + x).

(3.294) (3.295)

Exercises 3.3.1 Use the series solution method to solve the Volterra integral equations of the ﬁrst kind: x x (x − t + 1)u(t)dt 2. x cosh x − sinh x = u(t)dt 1. ex − 1 − x = 0

3. 1 + xex − ex = 5. − 1 − x +

0

x 0

1 4. 1 + x3 + xex − ex = 3

tu(t)dt

1 3 x + ex = 6

x

0

8. x − 2 sinh x + x cosh x = 9. 1 + x − sin x − cos x =

x 0 x

0 x

0

x

0

tu(t)dt

(x − t)u(t)dt

6. − x + 2 sin x − x cos x =

(x − t)u(t)dt

7. − 1 + cosh x =

x 0

(x − t)u(t)dt

(x − t)u(t)dt 10. 1 − x − e−x =

(x − t)u(t)dt

1 2 x + ln(1 + x) + x ln(1 + x) = 2 x 1 12. x2 ex = ex−t u(t)dt 2 0

11. − x +

x

0

(x − t)u(t)dt

x 0

u(t)dt

3.3.2 The Laplace Transform Method The Laplace transform method is a powerful technique that we used before for solving initial value problems and Volterra integral equations of the second kind. In the convolution theorem for the Laplace transform method, it was stated that if the kernel K(x, t) of the integral equation x K(x, t)u(t)dt, (3.296) f (x) = 0

depends on the diﬀerence x − t, then it is called a diﬀerence kernel. The Volterra integral equation of the ﬁrst kind can thus be expressed as x f (x) = K(x − t)u(t)dt. (3.297) 0

Consider two functions f1 (x) and f2 (x) that possess the conditions needed for the existence of Laplace transform for each. Let the Laplace transforms for the functions f1 (x) and f2 (x) be given by

112

3 Volterra Integral Equations

L{f1 (x)} = F1 (s), (3.298) L{f2 (x)} = F2 (s). The Laplace convolution product of these two functions is deﬁned by x (f1 ∗ f2 )(x) = f1 (x − t)f2 (t)dt, (3.299) 0

or (f2 ∗ f1 )(x) =

x

0

f2 (x − t)f1 (t)dt.

(3.300)

Recall that (f1 ∗ f2 )(x) = (f2 ∗ f1 )(x).

(3.301)

We can easily show that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x f1 (x − t)f2 (t)dt = F1 (s)F2 (s). (3.302) L{(f1 ∗ f2 )(x)} = L 0

Based on this summary, we will examine speciﬁc Volterra integral equations of the ﬁrst kind where the kernel is a diﬀerence kernel. Recall that we will apply the Laplace transform method and the inverse of the Laplace transform using Table 1.1 in Section 1.5. By taking Laplace transform of both sides of (3.297) we ﬁnd F (s) = K(s)U (s), (3.303) where U (s) = L{u(x)}, Solving (3.303) gives

K(s) = L{K(x)}, U (s) =

F (s) = L{f (x)}.

F (s) , K(s)

(3.304) (3.305)

where K(s) = 0.

(3.306)

The solution u(x) is obtained by taking the inverse Laplace transform of both sides of (3.305) where we ﬁnd F (s) . (3.307) u(x) = L−1 K(s) Recall that the right side of (3.307) can be evaluated by using Table 1.1 in Section 1.5. The Laplace transform method for solving Volterra integral equations of the ﬁrst kind will be illustrated by studying the following examples. Example 3.34 Solve the Volterra integral equation of the ﬁrst kind by using the Laplace transform method x x e − sin x − cos x = 2ex−tu(t)dt. (3.308) 0

Taking the Laplace transform of both sides of (3.308) yields

3.3 Volterra Integral Equations of the First Kind

113

1 s 2 1 − − = U (s), s − 1 s2 + 1 s2 + 1 s−1

(3.309)

2 2 = U (s). (s − 1)(s2 + 1) s−1

(3.310)

or equivalently

This in turn gives 1 . (3.311) +1 Taking the inverse Laplace transform of both sides gives the exact solution U (s) =

s2

u(x) = sin x.

(3.312)

Example 3.35 Solve the Volterra integral equation of the ﬁrst kind by using the Laplace transform method x x 1+x−e = (t − x)u(t)dt. (3.313) 0

Notice that the kernel is (t − x) = −(x − t). Taking the Laplace transform of both sides of (3.313) yields 1 1 1 1 + = − 2 U (s), − s s2 s−1 s so that 1 s(s − 1) + s − 1 − s2 = − 2 U (s). 2 s (s − 1) s

(3.314)

(3.315)

Solving for U (s) we ﬁnd 1 . (3.316) s−1 Taking the inverse Laplace transform of both sides gives the exact solution U (s) =

u(x) = ex .

(3.317)

Example 3.36 Solve the Volterra integral equation of the ﬁrst kind by using the Laplace transform method x 1 −1 + x2 + x3 + 2 sinh x + cosh x = (x − t + 2)u(t)dt. (3.318) 6 0 Taking the Laplace transform of both sides of (3.318) yields s3 + s2 − 1 1 2 = U (s), (3.319) + s2 (s2 − 1) s2 s or equivalently 1 s . (3.320) U (s) = 2 + 2 s s −1 Taking the inverse Laplace transform of both sides gives the exact solution u(x) = x + cosh x. (3.321)

114

3 Volterra Integral Equations

Exercises 3.3.2 Use the Laplace transform method to solve the Volterra integral equations of the ﬁrst kind: x x 1. x − sin x = (x − t)u(t)dt 2. ex + sin x − cos x = 2ex−t u(t)dt 0

0

x 1 3. 1 + x3 − cos x = (x − t)u(t)dt 3! 0 x (1 + 2(x − t))u(t)dt 5. x = 0 x

7. x =

0

9. 1 + x −

6. sinh x =

1 3 x − ex = 3!

x

0

x

0

x

0

(x − t)u(t)dt

ex−t u(t)dt

8. 1 − x − e−x =

(x − t + 1)u(t)dt

4. 1 + x − sin x − cos x =

x

0

(t − x)u(t)dt

(t − x)u(t)dt

x 1 2 1 x + x3 − sin x − cos x = (x − t + 1)u(t)dt 2 3! 0 x 11. 3 − 7x + x2 + sinh x − 3 cosh x = (x − t − 3)u(t)dt 10. 1 + x +

12. 1 − cos x =

0

x

0

cos(x − t)u(t)dt

3.3.3 Conversion to a Volterra Equation of the Second Kind In this section we will present a method that will convert Volterra integral equations of the ﬁrst kind to Volterra integral equations of the second kind. The conversion technique works eﬀectively only if K(x, x) = 0. Diﬀerentiating both sides of the Volterra integral equation of the ﬁrst kind x K(x, t)u(t)dt, (3.322) f (x) = 0

with respect to x, and using Leibnitz rule, we ﬁnd x Kx (x, t)u(t)dt. f (x) = K(x, x)u(x) + 0

(3.323)

Solving for u(x), provided that K(x, x) = 0, we obtain the Volterra integral equation of the second kind given by x f (x) 1 u(x) = (3.324) − Kx (x, t)u(t)dt. K(x, x) K(x, x) 0 Notice that the non-homogeneous term and the kernel have changed to f (x) g(x) = , (3.325) K(x, x) and

3.3 Volterra Integral Equations of the First Kind

G(x, t) = −

115

1 Kx (x, t), K(x, x)

(3.326)

respectively. Having converted the Volterra integral equation of the ﬁrst kind to the Volterra integral equation of the second kind, we then can use any method that was presented before. Because we solved the Volterra integral equations of the second kind by many methods, we will select distinct methods for solving the Volterra integral equation of the ﬁrst kind after reducing it to a second kind Volterra integral equation. Example 3.37 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x ex−t u(t)dt. (3.327) sinh x = 0

Diﬀerentiating both sides of (3.327) and using Leibnitz rule we obtain x cosh x = u(x) + ex−tu(t)dt, (3.328) 0

that gives the Volterra integral equation of the second kind x ex−tu(t)dt. u(x) = cosh x −

(3.329)

0

We select the Laplace transform method for solving this problem. Taking Laplace transform of both sides gives s 1 U (s) = 2 − U (s), (3.330) s −1 s−1 that leads to 1 . (3.331) U (s) = s+1 Taking the inverse Laplace transform of both sides gives the exact solution u(x) = e−x . (3.332) Example 3.38 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x (x − t + 1)u(t)dt. (3.333) 1 + sin x − cos x = 0

Diﬀerentiating both sides of (3.333) and using Leibnitz rule we obtain the Volterra integral equation of the second kind x u(t)dt. (3.334) u(x) = cos x + sin x − 0

We select the modiﬁed decomposition method for solving this problem. Therefore we set the modiﬁed recurrence relation

116

3 Volterra Integral Equations

u0 (x) = cos x,

u1 (x) = sin x − uk+1 (x) = −

0

0

x

u0 (t)dt = 0,

(3.335)

x

uk (t)dt = 0, k 1.

This gives the exact solution by u(x) = cos x.

(3.336)

Example 3.39 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x 9x2 + 5x3 = (10x − 10t + 6)u(t)dt. (3.337) 0

Diﬀerentiating both sides of (3.337) and using Leibnitz rule we obtain x 10u(t)dt, (3.338) 18x + 15x2 = 6u(x) + 0

that gives the Volterra integral equation of the second kind 5 5 x u(x) = 3x + x2 − u(t)dt. (3.339) 2 3 0 We select the Adomian decomposition method combined with the noise terms phenomenon for solving this problem. Therefore we set the recurrence relation 5 5 x uk (t)dt, k 0, (3.340) u0 (x) = 3x + x2 , uk+1 (x) = − 2 3 0 that gives 5 5 25 u0 (x) = 3x + x2 , u1 (x) = − x2 − x3 . (3.341) 2 2 18 Canceling the noise term 52 x2 , that appear in u0 (x) and u1 (x), from u0 (x) gives the exact solution by u(x) = 3x. (3.342) Remarks 1. It was stated before that if K(x, x) = 0, then the conversion of the ﬁrst kind to the second kind fails. However, if K(x, x) = 0 and Kx (x, x) = 0, then by diﬀerentiating the Volterra integral equation of the ﬁrst kind as many times as needed, provided that K(x, t) is diﬀerentiable, then the equation will be reduced to the Volterra integral equation of the second kind. 2. The function f (x) must satisfy speciﬁc conditions to guarantee a unique continuous solution for u(x). The determination of these special conditions will be left as an exercise.

3.3 Volterra Integral Equations of the First Kind

117

However, for the ﬁrst remark, where K(x, x) = 0 but Kx (x, x) = 0, we will diﬀerentiate twice, by using Leibnitz rule, as will be shown by the following illustrative example. Example 3.40 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x x sinh x = 2 sinh(x − t)u(t)dt. (3.343) 0

Diﬀerentiating both sides of (3.343) and using Leibnitz rule we obtain x x cosh x + sinh x = 2 cosh(x − t)u(t)dt, (3.344) 0

which is still a Volterra integral equation of the ﬁrst kind. However, because Kx (x, x) = 0, we diﬀerentiate again to obtain the Volterra integral equation of the second kind x 1 sinh(x − t)u(t)dt. (3.345) u(x) = cosh x + x sinh x − 2 0 We select the modiﬁed decomposition method for solving this problem. Therefore we set the modiﬁed recurrence relation u0 (x) = cosh x, x 1 sinh(x − t)u0 (t)dt = 0, u1 (x) = x sinh x − 2 (3.346) 0 x uk+1 (x) = − sinh(x − t)uk (t)dt = 0, k 1. 0

The exact solution is given by u(x) = cosh x.

(3.347)

Exercises 3.3.3 In Exercises 1–12, use Leibnitz rule to convert the Volterra integral equation of the ﬁrst kind to a second kind and solve the resulting equation: x x 1. ex + sin x − cos x = 2ex−t u(t)dt 2. ex − cos x = ex−t u(t)dt 3. x =

0

x

0

(x − t + 1)u(t)dt

5. ex − x − 1 = 7. ex − 1 =

x 0

9. 5x4 + x5 =

x 0

(x − t + 1)u(t)dt

(x − t + 1)u(t)dt

x 0

(x − t + 1)u(t)dt

0

4. ex + sin x − cos x = 6.

1 2 x x e = 2

x

0

x 0

2 cos(x − t)u(t)dt

ex−t u(t)dt

8. sin x − cos x + 1 =

x 0

(x − t + 1)u(t)dt

118

3 Volterra Integral Equations

10. 4 + x − 4ex + 3xex =

x

0

(x − t + 2)u(t)dt

x 1 3 x + 3ex = (x − t + 2)u(t)dt 3! 0 x π 12. tan x − ln cos x = (x − t + 1)u(t)dt, x < 2 0 11. − 3 − x + x2 +

In Exercises 13–16, use Leibnitz rule twice to convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation: x x 13. x sin x = 2 sin(x − t)u(t)dt 14. ex − sin x − cos x = 2 sin(x − t)u(t)dt 0

15. sin x − cos x + e

−x

16. sin x − x cos x =

0

x

= 0 x

0

2 sin(x − t)u(t)dt

2 sinh(x − t)u(t)dt

References 1. M. Bocher, Integral Equations, Cambridge University Press, London, (1974). 2. P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, (1985). 3. B.L. Moiseiwitsch, Integral Equations, Longman, London and New York, (1977). 4. D. Porter and D.S. Stirling, Integral Equations: A Practical Treatment from Spectral Theory to Applications, Cambridge, (2004). 5. G. Adomian, Solving Frontier Problems of Physics: The decomposition method, Kluwer, (1994). 6. G. Adomian and R. Rach, Noise terms in decomposition series solution, Comput. Math. Appl., 24 (1992) 61–64. 7. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 8. A.M.Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 9. A.M. Wazwaz, A reliable modiﬁcation of the Adomian decomposition method, Appl. Math. Comput., 102 (1999) 77–86. 10. A.M. Wazwaz, Necessary conditions for the appearance of noise terms in decomposition solution series, Appl. Math. Comput., 81 (1997) 265–274. 11. J.H. He, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, 20 (2006) 1141–1199. 12. J.H. He, Variational iteration method for autonomous ordinary diﬀerential systems, Appl. Math. Comput., 114(2/3) (2000) 115–123.

Chapter 4

Fredholm Integral Equations

4.1 Introduction It was stated in Chapter 2 that Fredholm integral equations arise in many scientiﬁc applications. It was also shown that Fredholm integral equations can be derived from boundary value problems. Erik Ivar Fredholm (1866– 1927) is best remembered for his work on integral equations and spectral theory. Fredholm was a Swedish mathematician who established the theory of integral equations and his 1903 paper in Acta Mathematica played a major role in the establishment of operator theory. As stated before, in Fredholm integral equations, the integral containing the unknown function u(x) is characterized by ﬁxed limits of integration in the form b

K(x, t)u(t)dt,

u(x) = f (x) + λ

(4.1)

a

where a and b are constants. For the ﬁrst kind Fredholm integral equations, the unknown function u(x) occurs only under the integral sign in the form b f (x) = K(x, t)u(t)dt. (4.2) a

However, Fredholm integral equations of the second kind, the unknown function u(x) occurs inside and outside the integral sign. The second kind is represented by the form b u(x) = f (x) + λ K(x, t)u(t)dt. (4.3) a

The kernel K(x, t) and the function f (x) are given real-valued functions [9], and λ is a parameter. When f (x) = 0, the equation is said to be homogeneous. In this chapter, we will mostly use degenerate or separable kernels. A degenerate or a separable kernel is a function that can be expressed as the sum of the product of two functions each depends only on one variable. Such a kernel can be expressed in the form A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

120

4 Fredholm Integral Equations

K(x, t) =

n

fi (x)gi (t).

(4.4)

i=1

Examples of separable kernels are x − t, (x − t)2 , 4xt, etc. In what follows we state, without proof, the Fredholm alternative theorem. Theorem 4.1 (Fredholm Alternative Theorem) If the homogeneous Fredholm integral equation b K(x, t)u(t)dt (4.5) u(x) = λ a

has only the trivial solution u(x) = 0, then the corresponding nonhomogeneous Fredholm equation b u(x) = f (x) + λ K(x, t)u(t)dt (4.6) a

has always a unique solution. This theorem is known by the Fredholm alternative theorem [1]. Theorem 4.2 (Unique Solution) If the kernel K(x, t) in Fredholm integral equation (4.1) is continuous, real valued function, bounded in the square a x b and a t b, and if f (x) is a continuous real valued function, then a necessary condition for the existence of a unique solution for Fredholm integral equation (4.1) is given by |λ|M (b − a) < 1,

(4.7)

|K(x, t)| M ∈ R.

(4.8)

where

On the contrary, if the necessary condition (4.7) does not hold, then a continuous solution may exist for Fredholm integral equation. To illustrate this, we consider the Fredholm integral equation 1 (3x + t)u(t)dt. (4.9) u(x) = −2 − 3x + 0

It is clear that λ = 1, |K(x, t)| 4 and (b − a) = 1. This gives |λ|M (b − a) = 4 < 1.

(4.10)

However, the Fredholm equation (4.9) has an exact solution given by u(x) = 6x. (4.11) A variety of analytic and numerical methods have been used to handle Fredholm integral equations. The direct computation method, the successive approximations method, and converting Fredholm equation to an equivalent boundary value problem are among many traditional methods that were commonly used. However, in this text we will apply the recently developed methods, namely, the Adomian decomposition method (ADM), the modiﬁed decomposition method (mADM), and the variational iteration method (VIM) to handle the Fredholm integral equations. Some of the traditional

4.2 Fredholm Integral Equations of the Second Kind

121

methods, namely, successive approximations method, and the direct computation method will be employed as well. The emphasis in this text will be on the use of these methods rather than proving theoretical concepts of convergence and existence. The theorems of uniqueness, existence, and convergence are important and can be found in the literature. The concern will be on the determination of the solution u(x) of the Fredholm integral equations of the ﬁrst kind and the second kind.

4.2 Fredholm Integral Equations of the Second Kind We will ﬁrst study Fredholm integral equations of the second kind given by b K(x, t)u(t)dt. (4.12) u(x) = f (x) + λ a

The unknown function u(x), that will be determined, occurs inside and outside the integral sign. The kernel K(x, t) and the function f (x) are given real-valued functions, and λ is a parameter. In what follows we will present the methods, new and traditional, that will be used to handle the Fredholm integral equations (4.12).

4.2.1 The Adomian Decomposition Method The Adomian decomposition method (ADM) was introduced and developed by George Adomian in [2–5] and was used before in Chapter 3. The Adomian method will be brieﬂy outlined. The Adomian decomposition method consists of decomposing the unknown function u(x) of any equation into a sum of an inﬁnite number of components deﬁned by the decomposition series ∞ u(x) = un (x), (4.13) n=0

or equivalently u(x) = u0 (x) + u1 (x) + u2 (x) + · · ·

(4.14)

where the components un(x), n 0 will be determined recurrently. The Adomian decomposition method concerns itself with ﬁnding the components u0 , u1 , u2 , . . . individually. As we have seen before, the determination of these components can be achieved in an easy way through a recurrence relation that usually involves simple integrals that can be easily evaluated. To establish the recurrence relation, we substitute (4.13) into the Fredholm integral equation (4.12) to obtain

122

4 Fredholm Integral Equations ∞

b

un (x) = f (x) + λ

K(x, t) a

n=0

or equivalently

∞

un (t) dt,

(4.15)

n=0

b

u0 (x) + u1 (x) + u2 (x) + · · · = f (x) + λ

K(x, t) [u0 (t) + u1 (t) + · · · ] dt. a

(4.16) The zeroth component u0 (x) is identiﬁed by all terms that are not included under the integral sign. This means that the components uj (x), j 0 of the unknown function u(x) are completely determined by setting the recurrence relation b K(x, t)un (t)dt, n 0, (4.17) u0 (x) = f (x), un+1 (x) = λ a

or equivalently u0 (x) = f (x), b K(x, t)u0 (t)dt, u1 (x) = λ a

b

u2 (x) = λ

K(x, t)u1 (t)dt,

(4.18)

a

u3 (x) = λ

b

K(x, t)u2 (t)dt, a

and so on for other components. In view of (4.18), the components u0 (x), u1 (x), u2 (x), u3 (x), . . . are completely determined. As a result, the solution u(x) of the Fredholm integral equation (4.12) is readily obtained in a series form by using the series assumption in (4.13). It is clearly seen that the decomposition method converted the integral equation into an elegant determination of computable components. It was formally shown that if an exact solution exists for the problem, then the obtained series converges very rapidly to that exact solution. The convergence concept of the decomposition series was thoroughly investigated by many researchers to conﬁrm the rapid convergence of the resulting series. However, for concrete problems, where a closed form solution is not obtainable, a truncated number of terms is usually used for numerical purposes. The more components we use the higher accuracy we obtain. Example 4.1 Solve the following Fredholm integral equation 1 u(x) = ex − x + x tu(t)dt.

(4.19)

0

The Adomian decomposition method assumes that the solution u(x) has a series form given in (4.13). Substituting the decomposition series (4.13) into

4.2 Fredholm Integral Equations of the Second Kind

both sides of (4.19) gives ∞ x un (x) = e − x + x

1

t

0

n=0

or equivalently

u0 (x) + u1 (x) + u2 (x) + · · · = e − x + x

un(t)dt,

(4.20)

n=0

1

x

∞

123

t [u0 (t) + u1 (t) + u2 (t) + · · · ] dt.

0

(4.21) We identify the zeroth component by all terms that are not included under the integral sign. Therefore, we obtain the following recurrence relation 1 x u0 (x) = e − x, uk+1 (x) = x tuk (t)dt, k 0. (4.22) 0

Consequently, we obtain u0 (x) = ex − x, 1 u1 (x) = x tu0 (t)dt = x 0

u2 (x) = x u3 (x) = x

tu1 (t)dt = x

0

1

tu2 (t)dt = x

0

1

1

1 0 1 0 1 0

t(et − t)dt =

2 x, 3

2 2 2 t dt = x, 3 9

(4.23)

2 2 2 t dt = x, 9 27

1

2 2 2 t dt = x, 27 81 0 0 and so on. Using (4.13) gives the series solution 2 1 1 1 x u(x) = e − x + x 1 + + + + ··· . (4.24) 3 3 9 27 Notice that the inﬁnite geometric series at the right side has a1 = 1, and the ratio r = 13 . The sum of the inﬁnite series is therefore given by 3 1 S= (4.25) 1 = . 2 1− 3 u4 (x) = x

tu3 (t)dt = x

The series solution (4.24) converges to the closed form solution u(x) = ex ,

(4.26)

obtained upon using (4.25) into (4.24). Example 4.2 Solve the following Fredholm integral equation π2 u(t)dt. u(x) = sin x − x + x

(4.27)

0

Proceeding as before, we substitute the decomposition series (4.13) into both sides of (4.27) to ﬁnd

124

4 Fredholm Integral Equations ∞

un (x) = sin x − x + x

0

n=0

or equivalently u0 (x)+ u1 (x)+ u2 (x)+ · · · = sin x− x+ x

π 2

0

π 2

∞

un (t)dt,

(4.28)

n=0

[u0 (t) + u1 (t) + · · · ] dt. (4.29)

We identify the zeroth component by all terms that are not included under the integral sign. Therefore, we obtain the following recurrence relation: π2 u0 (x) = sin x − x, uk+1 (x) = x uk (t)dt, k 0. (4.30) 0

Consequently, we obtain u0 (x) = sin x − x, π2 π2 x, u0 (t)dt = x − u1 (x) = x 8 0 π2 π2 π4 u1 (t)dt = u2 (x) = x x− x, 8 64 (4.31) 0 π2 4 6 π π x− x, u2 (t)dt = u3 (x) = x 64 512 0 π2 π8 π6 x− x, u3 (t)dt = u4 (x) = x 512 4096 0 and so on. Using (4.13) gives the series solution 2 π2 π π4 u(x) = sin x − x + 1 − x+ − x 8 8 64 6 4 π6 π π8 π − x+ − x + ··· . (4.32) + 64 512 512 4096 We can easily observe the appearance of the noise terms, i.e the identical terms with opposite signs. Canceling these noise terms in (4.32) gives the exact solution u(x) = sin x. (4.33) Example 4.3 Solve the following Fredholm integral equation 1 4 x tu(t)dt. (4.34) u(x) = x + e − + 3 0 Substituting the decomposition series (4.13) into both sides of (4.34) gives 1 ∞ ∞ 4 x un (x) = x + e − + t un (t)dt, (4.35) 3 0 n=0 n=0 or equivalently

4.2 Fredholm Integral Equations of the Second Kind

125

1 4 u0 (x)+u1 (x)+u2 (x)+· · · = x+ex − + t [u0 (t) + u1 (t) + · · · ] dt. (4.36) 3 0 Proceeding as before, we set the following recurrence relation 1 4 x tuk (t)dt, k 0. (4.37) u0 (x) = x + e − , uk+1 (x) = 3 0 Consequently, we obtain 1 4 2 tu0 (t)dt = , u0 (x) = x + ex − , u1 (x) = 3 3 0 1 1 1 1 (4.38) u3 (x) = tu1 (t)dt = , tu2 (t)dt = , u2 (x) = 3 6 0 0 1 1 u4 (x) = , tu3 (t)dt = 12 0 and so on. Using (4.13) gives the series solution 1 1 1 4 2 x 1 + + + + ··· . (4.39) u(x) = x + e − + 3 3 2 4 8 Notice that the inﬁnite geometric series at the right side has a1 = 1, and the ratio r = 12 . The sum of the inﬁnite series is therefore given by 1 = 2. (4.40) S= 1 − 12 The series solution (4.39) converges to the closed form solution u(x) = x + ex . (4.41) Example 4.4 Solve the following Fredholm integral equation π tu(t)dt. u(x) = 2 + cos x +

(4.42)

0

Proceeding as before we ﬁnd ∞ un (x) = 2 + cos x +

u0 (x) + u1 (x) + u2 (x) + · · · = 2 + cos x +

t

0

n=0

or equivalently

π

0

∞

un (t)dt,

π

t [u0 (t) + u1 (t) + · · · ] dt. (4.44)

We next set the following recurrence relation π u0 (x) = 2 + cos x, uk+1 (x) = tuk (t)dt, 0

This in turn gives

(4.43)

n=0

k 0.

(4.45)

126

4 Fredholm Integral Equations

u0 (x) = 2 + cos x, π u0 (t)dt = −2 + π 2 , u1 (x) = u2 (x) = u3 (x) =

0

π

1 u1 (t)dt = −π 2 + π 4 , 2

π

1 1 u2 (t)dt = − π 4 + π 6 , 2 4

0

0

(4.46)

π

1 1 u3 (t)dt = − π 6 + π 8 , 4 8 0 and so on. Using (4.13) gives the series solution 1 4 2 2 u(x) = 2 + cos x + (−2 + π ) + −π + π 2 1 1 1 1 (4.47) + − π4 + π6 + − π6 + π8 + · · · . 2 4 4 8 We can easily observe the appearance of the noise terms, i.e the identical terms with opposite signs. Canceling these noise terms in (4.47) gives the exact solution u(x) = cos x. (4.48) u4 (x) =

Example 4.5 Solve the following Fredholm integral equation π 1 4 sec2 xu(t)dt. (4.49) u(x) = 1 + 2 0 Substituting the decomposition series (4.13) into both sides of (4.49) gives π4 ∞ ∞ 1 un (x) = 1 + sec2 x un (t)dt, (4.50) 2 0 n=0 n=0 or equivalently

π4 1 [u0 (t) + u1 (t) + · · · ] dt. (4.51) u0 (x)+ u1 (x)+ u2 (x)+ · · · = 1 + sec2 x 2 0 Proceeding as before, we set the recurrence relation π4 1 2 u0 (x) = 1, uk+1 (x) = sec x uk (t)dt, k 0. (4.52) 2 0 Consequently, we obtain u0 (x) = 1, π4 1 π 2 u0 (t)dt = sec2 x, u1 (x) = sec x 2 8 0

4.2 Fredholm Integral Equations of the Second Kind

u2 (x) =

1 sec2 x 2

1 u3 (x) = sec2 x 2 1 u4 (x) = sec2 x 2

π 4

0 π 4

0 π 4

0

127

u1 (t)dt =

π sec2 x, 16

u2 (t)dt =

π sec2 x, 32

u3 (t)dt =

π sec2 x, 64

(4.53)

and so on. Using (4.13) gives the series solution 1 1 1 π 2 u(x) = 1 + sec x 1 + + + + · · · . (4.54) 8 2 4 8 The sum of the inﬁnite series at the right side is S = 2. The series solution (4.54) converges to the closed form solution π (4.55) u(x) = 1 + sec2 x. 4 Example 4.6 Solve the following Fredholm integral equation π u(x) = πx + sin 2x + x tu(t)dt.

(4.56)

−π

Proceeding as before we ﬁnd ∞ un (x) = πx + sin 2x + x n=0

π

t −π

∞

un (t)dt.

(4.57)

n=0

To determine the components of u(x), we use the recurrence relation π tuk (t)dt, k 0. (4.58) u0 (x) = πx + sin 2x, uk+1 (x) = x −π

This in turn gives u0 (x) = πx + sin 2x, π 2 u0 (t)dt = −πx + π 4 x, u1 (x) = x 3 0 π 2 4 u2 (x) = x u1 (t)dt = − π 4 x + π 7 x, 3 9 0 π 4 8 u2 (t)dt = − π 7 x + π 10 x, u3 (x) = x 9 27 0 and so on. Using (4.13) gives the series solution 2 4 2 u(x) = πx + sin 2x + −π + π 4 x + − π 4 + π7 x 3 3 9 8 4 + − π 7 + π10 x + · · · . 9 27 Canceling the noise terms in (4.60) gives the exact solution u(x) = sin 2x.

(4.59)

(4.60)

(4.61)

128

4 Fredholm Integral Equations

Exercises 4.2.1 In Exercises 1–20, solve the following Fredholm integral equations by using the Adomian decomposition method 1 1 1. u(x) = ex + 1 − e + u(t)dt 2. u(x) = ex + e−1 u(t)dt 3. u(x) = cos x + 2x + 5. u(x) = ex+2 − 2

0

0

xtu(t)dt

0

1

ex+t u(t)dt

0

7. u(x) = x + (1 − x)ex + 8. u(x) = 1 +

1 sin2 x 2

9. u(x) = xex −

π 2

1 1 + 2 2

0

u(t)dt

1 2

19 2 x + 15

π 2

0

16. u(x) = −x + sin x +

6. u(x) = ex +

π 2

0

xtu(t)dt

ex+1 − 1 − x+1

1

0

ext u(t)dt

x2 et(x−1) u(t)dt

1

0

13. u(x) = x + sin x −

1

4. u(x) = sin x − x +

u(t)dt

0

11. u(x) = x cos x + 1 +

15. u(x) = 1 −

π

12. u(x) = sin x + 14. u(x) = 1 −

1 1 + 2 2

0

π 2

0

π 2

u(t)dt

sin x cos tu(t)dt

1 2 x + 15

1 −1

(xt + x2 t2 )u(t)dt

(xt + x2 t2 )u(t)dt

π 2

0

u(t)dt

xu(t)dt

−1

π

0

1

10. u(x) = x sin x −

(1 + x − t)u(t)dt

1 e−(x+1) − 1 e−xt u(t)dt + x+1 0 1 3 1 18. u(x) = ex − ex+2 + ex+t u(t)dt 2 2 0 π π π 1 2 4 cos x sin tu(t)dt 20. u(x) = − sec2 x − u(t)dt 19. u(x) = cos x + 2 4 0 0

17. u(x) = e−x +

4.2.2 The Modiﬁed Decomposition Method As stated before, the Adomian decomposition method provides the solutions in an inﬁnite series of components. The components uj , j 0 are easily computed if the inhomogeneous term f (x) in the Fredholm integral equation b u(x) = f (x) + λ K(x, t)u(t)dt, (4.62) a

consists of a polynomial of one or two terms. However, if the function f (x) consists of a combination of two or more of polynomials, trigonometric func-

4.2 Fredholm Integral Equations of the Second Kind

129

tions, hyperbolic functions, and others, the evaluation of the components uj , j 0 requires more work. A reliable modiﬁcation of the Adomian decomposition method was presented and used in Chapter 3, and it was shown that this modiﬁcation facilitates the computational work and accelerates the convergence of the series solution. As presented before, the modiﬁed decomposition method depends mainly on splitting the function f (x) into two parts, therefore it cannot be used if the f (x) consists of only one term. The modiﬁed decomposition method will be brieﬂy outlined here, but will be used in this section and in other chapters as well. The standard Adomian decomposition method employs the recurrence relation u0 (x) = f (x), b (4.63) uk+1 (x) = λ K(x, t)uk (t)dt, k 0, a

where the solution u(x) is expressed by an inﬁnite sum of components deﬁned by ∞ un (x). (4.64) u(x) = n=0

In view of (4.63), the components un (x), n 0 are readily obtained. The modiﬁed decomposition method presents a slight variation to the recurrence relation (4.63) to determine the components of u(x) in an easier and faster manner. For many cases, the function f (x) can be set as the sum of two partial functions, namely f1 (x) and f2 (x). In other words, we can set (4.65) f (x) = f1 (x) + f2 (x). In view of (4.65), we introduce a qualitative change in the formation of the recurrence relation (4.63). The modiﬁed decomposition method identiﬁes the zeroth component u0 (x) by one part of f (x), namely f1 (x) or f2 (x). The other part of f (x) can be added to the component u1 (x) that exists in the standard recurrence relation. The modiﬁed decomposition method admits the use of the modiﬁed recurrence relation: u0 (x) = f1 (x), b K(x, t)u0 (t)dt, u1 (x) = f2 (x) + λ (4.66) a b uk+1 (x) = λ K(x, t)uk (t)dt, k 1. a

It is obvious that the diﬀerence between the standard recurrence relation (4.63) and the modiﬁed recurrence relation (4.66) rests only in the formation of the ﬁrst two components u0 (x) and u1 (x) only. The other components uj , j 2 remain the same in the two recurrence relations. Although this variation in the formation of u0 (x) and u1 (x) is slight, however it has been shown that it accelerates the convergence of the solution and minimizes the

130

4 Fredholm Integral Equations

size of calculations. Moreover, reducing the number of terms in f1 (x) aﬀects not only the component u1 (x), but also the other components as well. We here emphasize on the two important remarks made in Chapter 3. First, by proper selection of the functions f1 (x) and f2 (x), the exact solution u(x) may be obtained by using very few iterations, and sometimes by evaluating only two components. The success of this modiﬁcation depends only on the proper choice of f1 (x) and f2 (x), and this can be made through trials only. A rule that may help for the proper choice of f1 (x) and f2 (x) could not be found yet. Second, if f (x) consists of one term only, the modiﬁed decomposition method cannot be used in this case. Example 4.7 Solve the Fredholm integral equation by using the modiﬁed decomposition method 1 1 4x 4 tu(t)dt. (4.67) u(x) = 3x + e − (17 + 3e ) + 16 0 We ﬁrst decompose f (x) given by f (x) = 3x + e4x −

1 (17 + 3e4 ), 16

(4.68)

into two parts, namely 1 (17 + 3e4 ). 16 We next use the modiﬁed recurrence formula (4.66) to obtain u0 (x) = f1 (x) = 3x + e4x , 1 1 tu0 (t)dt = 0, u1 (x) = − (17 + 3e4 ) + 16 0 x uk+1 (x) = K(x, t)uk (t)dt = 0, k 1. f1 (x) = 3x + e4x ,

f2 (x) = −

(4.69)

(4.70)

0

It is obvious that each component of uj , j 1 is zero. This in turn gives the exact solution by u(x) = 3x + e4x . (4.71) Example 4.8 Solve the Fredholm integral equation by using the modiﬁed decomposition method 1 π 1 − 2 sinh earctan t u(t)dt. (4.72) u(x) = + 1 + x2 4 −1 Proceeding as before we split f (x) given by 1 π f (x) = − 2 sinh , 2 1+x 4 into two parts, namely

(4.73)

4.2 Fredholm Integral Equations of the Second Kind

1 π , f2 (x) = −2 sinh . 1 + x2 4 We next use the modiﬁed recurrence formula (4.66) to obtain 1 , u0 (x) = f1 (x) = 1 + x2 1 π u1 (x) = −2 sinh + earctan t u0 (t)dt = 0, 4 −1 1 earctan t uk (t)dt = 0, k 1. uk+1 (x) = f1 (x) =

131

(4.74)

(4.75)

−1

It is obvious that each component of uj , j 1 is zero. This in turn gives the exact solution by 1 u(x) = . (4.76) 1 + x2 Example 4.9 Solve the Fredholm integral equation by using the modiﬁed decomposition method 2−π 2 1 2 1 −1 x + 1 u(t)dt. (4.77) u(x) = x + sin + x + x 2 2 2 −1 We decompose f (x) given by f (x) = x + sin−1

x+1 2−π 2 + x , 2 2

(4.78)

into two parts given by 2−π 2 x+1 , f2 (x) = x . 2 2 We next use the modiﬁed recurrence formula (4.66) to obtain x+1 u0 (x) = x + sin−1 , 2 2−π 2 1 2 1 x + x u0 (t)dt = 0, u1 (x) = 2 2 −1 1 uk (t)dt = 0, k 1. uk+1 (x) = − f1 (x) = x + sin−1

(4.79)

(4.80)

−1

It is obvious that each component of uj , j 1 is zero. The exact solution is therefore given by x+1 . (4.81) u(x) = x + sin−1 2 Example 4.10 Solve the Fredholm integral equation by using the modiﬁed decomposition method

132

4 Fredholm Integral Equations

u(x) = sec2 x + x2 + x −

π 4

0

4 2 x + x u(t) dt. π

(4.82)

The function f (x) consists of three terms. By trial we split f (x) given by f (x) = sec2 x + x2 + x, into two parts

f1 (x) = sec2 x,

f2 (x) = x2 + x.

Using the modiﬁed recurrence formula (4.66) gives u0 (x) = f1 (x) = sec2 x, π4 4 2 2 x + xu0 (t) dt = 0, u1 (x) = x + x − π 0 π4 K(x, t)uk (t)dt = 0, k 1. uk+1 (x) =

(4.83) (4.84)

(4.85)

0

As a result, the exact solution is given by u(x) = sec2 x.

(4.86)

Exercises 4.2.2 Use the modiﬁed decomposition method to solve the following Fredholm integral equations: π 2 1. u(x) = sin x − x + x tu(t)dt 0

2. u(x) = (π + 2)x + sin x − cos x − x 3. u(x) = ex + 12x2 + (3 + e)x − 4 −

π

tu(t)dt

0 1

0

(x − t)u(t)dt

1 x+1 x−1 − sin−1 − xu(t)dt 2 2 0 1 (x4 − t4 )u(t)dt 5. u(x) = −6 + 14x4 + 21x2 + x − 4. u(x) = (π − 2)x + sin−1

−1

6. u(x) = x + x4 + 9ex+1 − 23ex − 7. u(x) = x + ex − 2ex−1 + 2ex − 8. u(x) = xex +

xex+1 + 1 − (x + 1)2

1 0

1

0

1

0

ex+t u(t)dt

ex−t u(t)dt

ext u(t)dt

9. u(x) = ex+1 + ex−1 + (e2 + 1)ex−1 − 10. u(x) =

7 2 + x + x2 + x3 − 15 12

1

0

1

0

ex−t u(t)dt

(1 + x − t)u(t)dt

4.2 Fredholm Integral Equations of the Second Kind 11. u(x) = e2x −

1 2 (e + 1)x + 4

1

12. u(x) = ex (e 3 − 1) + e2x − 13. u(x) = x + ex − xex +

1

0

1 0

1 3

133

xtu(t)dt

1

0

5

ex− 3 t u(t)dt

x2 et(x−1) u(t)dt

14. u(x) = (π 2 + 2π − 4) − (π + 2)x + x(sin x − cos x) +

π 0

(x − t)u(t)dt

1 (π − 2)(x + 1) + x tan−1 x − (1 + x − t)u(t)dt 2 −1 1 1+e ex − xu(t)dt 16. u(x) = ln x+ 2 1 + ex 0 15. u(x) =

4.2.3 The Noise Terms Phenomenon It was shown that a proper selection of f1 (x) and f2 (x) is essential to use the modiﬁed decomposition method. However, the noise terms phenomenon, that was introduced in Chapter 3, demonstrated a fast convergence of the solution. This phenomenon was presented before, therefore we present here the main steps for using this eﬀect concept. The noise terms as deﬁned before are the identical terms with opposite signs that may appear between components u0 (x) and u1 (x). Other noise terms may appear between other components. By canceling the noise terms between u0 (x) and u1 (x), even though u1 (x) contains further terms, the remaining non-canceled terms of u0 (x) may give the exact solution of the integral equation. The appearance of the noise terms between u0 (x) and u1 (x) is not always suﬃcient to obtain the exact solution by canceling these noise terms. Therefore, it is necessary to show that the non-canceled terms of u0 (x) satisfy the given integral equation. It was formally proved in [6] that a necessary condition for the appearance of the noise terms is required. The conclusion made in [6] is that the zeroth component u0 (x) must contain the exact solution u(x) among other terms. The phenomenon of the useful noise terms will be explained by the following illustrative examples. Example 4.11 Solve the Fredholm integral equation by using the noise terms phenomenon: π2 u(x) = x sin x − x + xu(t)dt. (4.87) 0

Following the standard Adomian method we set the recurrence relation

134

4 Fredholm Integral Equations

u0 (x) = x sin x − x, π2 xuk (t)dt, uk+1 (x) = 0

This gives

k 0.

u0 (x) = x sin x − x, π2 π2 x. xu0 (t)dt = x − u1 (x) = 8 0

(4.88)

(4.89)

The noise terms ∓x appear in u0 (x) and u1 (x). Canceling this term from the zeroth component u0 (x) gives the exact solution u(x) = x sin x, (4.90) that justiﬁes the integral equation. The other terms of u1 (x) vanish in the limit with other terms of the other components. Example 4.12 Solve the Fredholm integral equation by using the noise terms phenomenon: π2 π u(x) = sin x + cos x − x + xtu(t)dt. (4.91) 2 0 The standard Adomian method gives the recurrence relation π u0 (x) = sin x + cos x − x, 2 π2 (4.92) xtuk (t)dt, k 0. uk+1 (x) = 0

This gives

π u0 (x) = sin x + cos x − x, 2 π2 π π4 xtu0 (t)dt = x − u1 (x) = x. 2 48 0

(4.93)

The noise terms ∓ π2 x appear in u0 (x) and u1 (x). Canceling this term from the zeroth component u0 (x) gives the exact solution u(x) = sin x + cos x, (4.94) that justiﬁes the integral equation. It is to be noted that the other terms of u1 (x) vanish in the limit with other terms of the other components. Example 4.13 Solve the Fredholm integral equation by using the noise terms phenomenon: π2 sin x π3 2 u(t)dt. (4.95) u(x) = x + + x + x ln 2 − x 1 + cos x 24 0 The standard Adomian method gives the recurrence relation:

4.2 Fredholm Integral Equations of the Second Kind

π3 sin x + x + x ln 2, u0 (x) = x2 + 1 + cos x 24 π2 u(t)dt, k 0. uk+1 (x) = −x

135

(4.96)

0

This gives sin x π3 u0 (x) = x2 + + x + x ln 2, 1 + cos x 24 π2 π2 ln 2 π5 π3 x− x. xu0 (t)dt = − x − x ln 2 − u1 (x) = − 24 8 192 0

(4.97)

3

The noise terms ± π24 x, ±x ln 2 appear in u0 (x) and u1 (x). Canceling these terms from the zeroth component u0 (x) gives the exact solution sin x u(x) = x2 + , (4.98) 1 + cos x that justiﬁes the integral equation. The other terms of u1 (x) vanish in the limit with other terms of the other components. Example 4.14 Solve the Fredholm integral equation by using the noise terms phenomenon π π3 2 u(x) = x + x cos x + x − 2x − x u(t)dt. (4.99) 3 0 Proceeding as before, we set the recurrence relation π3 x − 2x, u0 (x) = x2 + x cos x + 3 π (4.100) uk (t)dt, k 0. uk+1 (x) = −x 0

This gives π3 u0 (x) = x2 + x cos x + x − 2x, 3 π π3 π5 u0 (t)dt = −x −2 − π 2 + u1 (x) = −x + . 3 6 0

(4.101)

3

The noise terms ± π3 x, ∓2x appear in u0 (x) and u1 (x). Canceling these terms from the zeroth component u0 (x) gives the exact solution u(x) = x2 + x cos x, (4.102) that justiﬁes the integral equation (4.99).

Exercises 4.2.3 Use the noise terms phenomenon to solve the following Fredholm integral equations: π π cos x 2 xu(t)dt + ln 2 x + − 1. u(x) = 1 + 2 1 + sin x 0

136

4 Fredholm Integral Equations

π π2 2. u(x) = x 1 + π + xu(t)dt + x sin x − 2 0 π π3 4 xu(t)dt 3. u(x) = x 1 + + x2 + sec2 x − 192 0 π xu(t)dt 4. u(x) = x(π − 2 + sin x + cos x) −

+ ln 2 +

0

π sin x 2 − xu(t)dt 8 1 + cos x 0 π cos x 2 xu(t)dt 6. u(x) = x(1 + ln 2) + sin x + − 1 + sin x 0 π sin x(2 + sin x) π 2 − 7. u(x) = x + xu(t)dt 2 1 + sin x 0 π sin x π 2 −1 x+ − 8. u(x) = xu(t)dt 2 1 + sin x 0 √ π π− 3 cos x 3 xu(t)dt 9. u(x) = x+ − 3 1 + sin x 0 π π2 2 xu(t)dt 10. u(x) = 4 − x + x2 (sin x + cos x) + 2 −π 2 π 1 4 xu(t)dt 11. u(x) = x + sin(2x) − 4 0 π π−2 4 xu(t)dt 12. u(x) = x + x cos(2x) − 8 0 1 π2 1 − x tan−1 tu(t)dt 13. u(x) = x + 1 + x2 32 0 1 14. u(x) = πx + cos−1 x − xu(t)dt 5. u(x) = x 1 +

π2

−1

1 π 15. u(x) = − x + x cos−1 x − xu(t)dt 4 −1 1 π−2 x + x tan−1 x − 16. u(x) = xu(t)dt 2 −1

4.2.4 The Variational Iteration Method In Chapter 3, the variational iteration method was used to handle Volterra integral equations, where the Volterra integral equation was converted to an initial value problem or to an equivalent integro-diﬀerential equation. The method provides rapidly convergent successive approximations to the exact solution if such a closed form solution exists. In this section, we will apply the variational iteration method to handle Fredholm integral equation. The method works eﬀectively if the kernel

4.2 Fredholm Integral Equations of the Second Kind

137

K(x, t) is separable and can be written in the form K(x, t) = g(x)h(t). The approach to be used here is identical to the approach used in the previous chapter. This means that we should diﬀerentiate both sides of the Fredholm integral equation to convert it to an identical Fredholm integro-diﬀerential equation. It is important to note that integro-diﬀerential equation needs an initial condition that should be deﬁned. In view of this fact, we will study only the cases where g(x) = xn , n 1. Solving Fredholm integro-diﬀerential equation by the variational iteration method will be studied again in details in Chapter 6. The standard Fredholm integral equation is of the form b K(x, t)u(t)dt, (4.103) u(x) = f (x) + a

or equivalently

b

u(x) = f (x) + g(x)

h(t)u(t)dt.

(4.104)

a

Recall that the integral at the right side represents a constant value. Diﬀerentiating both sides of (4.104) with respect to x gives b u (x) = f (x) + g (x) h(t)u(t)dt. (4.105) a

The correction functional for the integro-diﬀerential equation (4.105) is x b λ(ξ) un (ξ) − f (ξ) − g (ξ) h(r)˜ un (r)dr dξ. un+1 (x) = un (x) + 0

a

(4.106) As presented before, the variational iteration method is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ(ξ) that can be identiﬁed optimally via integration by parts and by using a restricted variation. However, λ(ξ) = −1 for ﬁrst order integro-diﬀerential equations. Having determined λ, an iteration formula, without restricted variation, given by x b h(r)un (r)dr dξ, (4.107) un+1 (x) = un (x) − un (ξ) − f (ξ) − g (ξ) 0

a

is used for the determination of the successive approximations un+1 (x), n 0 of the solution u(x). The zeroth approximation u0 can be any selective function. However, using the given initial value u(0) is preferably used for the selective zeroth approximation u0 as will be seen later. Consequently, the solution is given by (4.108) u(x) = lim un (x). n→∞

138

4 Fredholm Integral Equations

The variational iteration method will be illustrated by studying the following Fredholm integral equations. Example 4.15 Use the variational iteration method to solve the Fredholm integral equation 1 u(x) = ex − x + x tu(t)dt. (4.109) 0

Diﬀerentiating both sides of this equation with respect to x yields 1 tu(t)dt. u (x) = ex − 1 +

(4.110)

The correction functional for this equation is given by x 1 ξ run (r)dr dξ, un+1 (x) = un (x) − un (ξ) − e + 1 −

(4.111)

0

0

0

where we used λ = −1 for ﬁrst-order integro-diﬀerential equations. Notice that the initial condition u(0) = 1 is obtained by substituting x = 0 into (4.109). We can use the initial condition to select u0 (x) = u(0) = 1. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1,

u0 (ξ) − eξ + 1 −

1 ru0 (r)dr dξ = ex − x, 2 0 0 x 1 1 x, u2 (x) = u1 (x) − ru1 (r)dr dξ = ex − u1 (ξ) − eξ + 1 − 2 × 3 0 0 x 1 1 ξ u3 (x) = u2 (x) − ru2 (r)dr dξ = ex − x, u2 (ξ) − e + 1 − 2 × 32 0 0 x 1 1 u3 (ξ) − eξ + 1 − u4 (x) = u3 (x) − ru3 (r)dr dξ = ex − x, 2 × 33 0 0 .. . 1 x, n 0. (4.112) un+1 = ex − 2 × 3n The VIM admits the use of u(x) = lim un (x) = ex . (4.113) u1 (x) = u0 (x) −

x

1

n→∞

Example 4.16 Use the variational iteration method to solve the Fredholm integral equation π2 u(t)dt. (4.114) u(x) = sin x − x + x 0

Diﬀerentiating both sides of this equation with respect to x gives

4.2 Fredholm Integral Equations of the Second Kind

u (x) = cos x − 1 +

π 2

139

u(t)dt.

(4.115)

0

The correction functional for this equation is given by x π2 un+1 (x) = un (x) − un (ξ) − cos ξ + 1 − un (r)dr dξ, 0

(4.116)

0

where we used λ = −1 for ﬁrst-order integro-diﬀerential equations. The initial condition u(0) = 0 is obtained by substituting x = 0 into (4.114). We can use the initial condition to select u0 (x) = u(0) = 0. Using this selection into the correction functional gives the following successive approximations u0 (x) = 0, x π2 u0 (ξ) − cos ξ + 1 − u1 (x) = u0 (x) − u0 (r)dr dξ = sin x − x, u2 (x) = u1 (x) −

0

0

x

u1 (ξ)

π 2

− cos ξ + 1 − u1 (r)dr dξ 0 π2 x , = (sin x − x) + x − 8 π 0

x

(4.117)

2

u2 (ξ) − cos ξ + 1 − u2 (r)dr dξ 0 0 2 π2 π π4 = (sin x − x) + x − x + x− x , 8 8 64 and so on. Canceling the noise terms, the exact solution is given by u3 (x) = u2 (x) −

u(x) = sin x.

(4.118)

Example 4.17 Use the variational iteration method to solve the Fredholm integral equation π xu(t)dt. (4.119) u(x) = −2x + sin x + cos x + 0

Diﬀerentiating both sides of this equation with respect to x gives π u (x) = −2 + cos x − sin x + u(t)dt.

(4.120)

0

The correction functional for this equation is given by x un (ξ) + 2 − cos ξ + sin ξ − un+1 (x) = un (x) − 0

0

π

un (r)dr dξ.

(4.121) The initial condition u(0) = 1 is obtained by substituting x = 0 into (4.119). Using this selection into the correction functional gives the following successive approximations

140

u0 (x) = 1, u1 (x) = u0 (x) −

4 Fredholm Integral Equations x

0

u0 (ξ)

+ 2 − cos ξ + sin ξ −

π

0

u0 (r)dr dξ

= sin x + cos x + (πx − 2x), x π u1 (r)dr dξ u1 (ξ) + 2 − cos ξ + sin ξ − u2 (x) = u1 (x) − 0 0 π3 x , (4.122) = sin x + cos x + (πx − 2x) + −πx + 2x − π 2 x + 2 x π u3 (x) = u2 (x) − u2 (r)dr dξ u2 (ξ) + 2 − cos ξ + sin ξ − 0 0 π3 x = sin x + cos x + (πx − 2x) + −πx + 2x − π 2 x + 2 3 π x + ··· , + π2 x − 2 and so on. Canceling the noise terms, the exact solution is given by u(x) = sin x + cos x. (4.123) Example 4.18 Use the variational iteration method to solve the Fredholm integral equation π2 u(x) = −x3 + cos x + x3 u(t)dt. (4.124) 0

Diﬀerentiating both sides of this equation with respect to x gives π2 u (x) = −3x2 − sin x + 3x2 u(t)dt.

(4.125)

0

The correction functional for this equation is given by π2 x 2 2 un (r)dr dξ. (4.126) un (ξ) + 3ξ + sin ξ − 3ξ un+1 (x) = un (x)− 0

0

The initial condition is u(0) = 1. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1, π x

2

u0 (r)dr dξ u0 (ξ) + 3ξ 2 + sin ξ − 3ξ 2 u1 (x) = u0 (x) − 0 0 π = cos x + x3 − x3 , 2 π x

2

u1 (r)dr dξ u1 (ξ) + 3ξ 2 + sin ξ − 3ξ 2 0 π π 3 π4 3 π5 3 3 3 3 = cos x + x −x + − x +x − x + x , 2 2 64 128 and so on. Canceling the noise terms, the exact solution is given by u2 (x) = u1 (x) −

0

(4.127)

4.2 Fredholm Integral Equations of the Second Kind

141

u(x) = cos x.

(4.128)

Exercises 4.2.4 Solve the following Fredholm integral equations by using the the variational iteration method π π 2 1. u(x) = cos x + 2x + xtu(t)dt 2. u(x) = sin x − x + xtu(t)dt 1 2 x + 3. u(x) = 1 − 15 4. u(x) = 1 −

19 2 x + 15

0

1

−1

1

−1

0

(xt + x2 t2 )u(t)dt (xt + x2 t2 )u(t)dt

5. u(x) = (π + 2)x + sin x − cos x − x 6. u(x) = e2x −

1 2 (e + 1)x + 4

7. u(x) = 1 + 9x + 2x2 + x3 −

π

0

tu(t)dt

1

xtu(t)dt

0

1 0

(20xt + 10x2 t2 )u(t)dt

π π x + sin x − cos x + x cos tu(t)dt 2 0 2 1 3 1 xtu(t)dt 10. u(x) = 2x + ex − xtu(t)dt 9. u(x) = 1 + x + ex − 3 0 4 0 π 1 1 4 + sin(2x) − 11. u(x) = x xu(t)dt 12. u(x) = ex + 2xe−1 − xtu(t)dt 4 −1 0 3 π π π3 π 2 xtu(t)dt 14. u(x) = xtu(t)dt + 3 x − cos x − x − sin x − 13. u(x) = 3 24 0 0 π √ 4 x sec tu(t)dt 15. u(x) = 2 x + sec x + tan x − 8. u(x) =

π 16. u(x) = x + sin x + cos x − 8

0

0

π 4

x sin tu(t)dt

4.2.5 The Direct Computation Method In this section, the direct computation method will be applied to solve the Fredholm integral equations. The method approaches Fredholm integral equations in a direct manner and gives the solution in an exact form and not in a series form. It is important to point out that this method will be applied for the degenerate or separable kernels of the form n gk (x)hk (t). (4.129) K(x, t) = k=1

142

4 Fredholm Integral Equations

Examples of separable kernels are x − t, xt, x2 − t2 , xt2 + x2 t, etc. The direct computation method can be applied as follows: 1. We ﬁrst substitute (4.129) into the Fredholm integral equation the form b u(x) = f (x) + K(x, t) u(t)dt. (4.130) a

2. This substitution gives b u(x) = f (x) + g1 (x) h1 (t)u(t)dt + g2 (x) a

b

h2 (t)u(t)dt + · · · a

b

+gn (x)

(4.131)

hn (t)u(t)dt. a

3. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (4.131) becomes u(x) = f (x) + λα1 g1 (x) + λα2 g2 (x) + · · · + λαn gn (x), where

αi =

(4.132)

b

hi (t)u(t)dt,

1 i n.

(4.133)

a

4. Substituting (4.132) into (4.133) gives a system of n algebraic equations that can be solved to determine the constants αi , 1 i n. Using the obtained numerical values of αi into (4.132), the solution u(x) of the Fredholm integral equation (4.130) is readily obtained. Example 4.19 Solve the Fredholm integral equation by using the direct computation method 1 1 2 2 x tu(t)dt. (4.134) u(x) = 3x + 3x + 2 0 The kernel K(x, t) = x2 t is separable. Consequently, we rewrite (4.134) as 1 2 1 2 u(x) = 3x + 3x + x tu(t)dt. (4.135) 2 0 The integral at the right side is equivalent to a constant because it depends only on functions of the variable t with constant limits of integration. Consequently, Equation (4.135) can be rewritten as 1 u(x) = 3x + 3x2 + αx2 , (4.136) 2 where 1 tu(t)dt. (4.137) α= 0

To determine α, we substitute (4.136) into (4.137) to obtain 1 1 α= t 3t + 3t2 + αt2 dt. 2 0

(4.138)

4.2 Fredholm Integral Equations of the Second Kind

143

Integrating the right side of (4.138) yields 7 1 α = + α, 4 8 that gives α = 2.

(4.139)

Substituting (4.140) into (4.136) leads to the exact solution u(x) = 3x + 4x2 , obtained by substituting α = 2 in (4.136).

(4.141)

(4.140)

Example 4.20 Solve the Fredholm integral equation by using the direct computation method π3 1 1 u(x) = x + sec x tan x − x u(t)dt. (4.142) 3 3 0 The integral at the right side is equivalent to a constant because it depends only on functions of the variable t with constant limits of integration. Consequently, Equation (4.142) can be rewritten as 1 1 (4.143) u(x) = x + sec x tan x − αx, 3 3 where π3 u(t)dt. (4.144) α= 0

To determine α, we substitute (4.143) into (4.144) to obtain π3 1 1 α= t + sec t tan t − αt dt. 3 3 0 Integrating the right side of (4.145) yields 1 1 α = 1 + π 2 − απ2 , 54 54 that gives α = 1.

(4.145)

(4.146) (4.147)

Substituting (4.147) into (4.143) gives the exact solution u(x) = sec x tan x.

(4.148)

Example 4.21 Solve the Fredholm integral equation by using the direct computation method 1 (30xt2 + 20x2 t)u(t)dt. (4.149) u(x) = 11x + 10x2 + x3 − 0

The kernel K(x, t) = 30xt2 + 20x2 t is separable. Consequently, we rewrite (4.149) as 1 1 u(x) = 11x + 10x2 + x3 − 30x t2 u(t)dt − 20x2 tu(t)dt. (4.150) 0

0

144

4 Fredholm Integral Equations

Each integral at the right side is equivalent to a constant because it depends only on functions of the variable t with constant limits of integration. Consequently, Equation (4.150) can be rewritten as u(x) = (11 − 30α)x + (10 − 20β)x2 + x3 , (4.151) where

1

α=

t2 u(t)dt,

0

β=

(4.152)

1

tu(t)dt. 0

To determine the constants α and β, we substitute (4.151) into (4.152) to obtain 1 59 15 t2 ((11 − 30α)x + (10 − 20β)x2 + x3 )dt = − α − 4β, α= 12 2 0 (4.153) 1 191 2 3 t((11 − 30α)t + (10 − 20β)t + x )dt = β= − 10α − 5β. 30 0 Solving this system of algebraic equations gives 9 11 , β= . (4.154) α= 30 20 Substituting (4.154) into (4.151) gives the exact solution u(x) = x2 + x3 .

(4.155)

Example 4.22 Solve the Fredholm integral equation by using the direct computation method 1 (1 + 30xt2 + 12x2 t)u(t)dt. (4.156) u(x) = 4 + 45x + 26x2 − 0

The kernel K(x, t) = 1 + 30xt2 + 12x2 t is separable. Consequently, we rewrite (4.156) as 1 1 1 u(t)dt − 30x t2 u(t)dt − 12x2 tu(t)dt. u(x) = 4 + 45x + 26x2 − 0

0

0

(4.157) Each integral at the right side is equivalent to a constant because it depends only on functions of the variable t with constant limits of integration. Consequently, Equation (4.157) can be rewritten as u(x) = (4 − α) + (45 − 30β)x + (26 − 12γ)x2 , (4.158) where

α=

1

u(t)dt, 0

β= 0

1

t2 u(t)dt,

1

γ=

tu(t)dt.

(4.159)

0

To determine the constants α, β and γ, we substitute (4.158) into each equation of (4.159) to obtain

4.2 Fredholm Integral Equations of the Second Kind

1

α=

145

0

(4 − α) + (45 − 30β)t + (26 − 12γ)t2 dt

211 = − α − 15β − 4γ, 6 1

β= t2 (4 − α) + (45 − 30β)t + (26 − 12γ)t2 dt 0

(4.160)

1067 1 15 12 = − α − β − γ, 60 3 2 5 1

γ= t (4 − α) + (45 − 30β)t + (26 − 12γ)t2 dt 0

47 1 = − α − 10β − 3γ. 2 2 Unlike the previous examples, we obtain a system of three equations in three unknowns α, β, and γ. Solving this system of algebraic equations gives 23 43 , γ= . (4.161) α = 3, β = 30 12 Substituting (4.161) into (4.158), the exact solution is given by (4.162) u(x) = 1 + 2x + 3x2 .

Exercises 4.2.5 Use the direct computation method to solve the following Fredholm integral equations: 1 (20xt + 10x2 t2 )u(t)dt 1. u(x) = 1 + 9x + 2x2 + x3 − 0

2. u(x) = −8 + 11x − x2 + x3 − 3. u(x) = −11 + 9x + x3 + x4 − 4. u(x) = −15 + 10x3 + x4 −

5. u(x) = 1 + 7x + 20x2 + x3 −

0

1 0

1

1

0

(12x − 20t)u(t)dt (20x − 30t)u(t)dt

(20x3 − 56t3 )u(t)dt

1 0

(10xt2 + 20x2 t)u(t)dt

π 2 6 √ − 1 x + sec x tan x − xu(t)dt 3 0 π √ 2π 3 − ln(2 + 3) x + sec x tan x − xtu(t)dt 7. u(x) = 3 0 1 ln(xt)u(t)dt, 0 < x 1 8. u(x) = 1 +

6. u(x) =

0+

9. u(x) = 1 + ln x −

1

0+

ln(xt2 )u(t)dt, 0 < x 1

146

4 Fredholm Integral Equations

1

x u(t)dt, 0 < x 1 t2 π tu(t)dt 11. u(x) = sin x + (π − 1) cos x − cos x

10. u(x) = 1 + ln x −

ln

0+

0

π π 12. u(x) = x + sin x − cos x + x cos tu(t)dt 2 0 π π π 4 4 13. u(x) = 1 + sec2 x − sec2 xu(t)dt 14. u(x) = 1 − sec2 xu(t)dt 2 0 0 2 1 3 1 xtu(t)dt 16. u(x) = 2x + ex − xtu(t)dt 15. u(x) = 1 + x + ex − 3 0 4 0

4.2.6 The Successive Approximations Method The successive approximations method, or the Picard iteration method was introduced before in Chapter 3. The method provides a scheme that can be used for solving initial value problems or integral equations. This method solves any problem by ﬁnding successive approximations to the solution by starting with an initial guess as u0 (x), called the zeroth approximation. As will be seen, the zeroth approximation is any selective real-valued function that will be used in a recurrence relation to determine the other approximations. The most commonly used values for the zeroth approximations are 0, 1, or x. Of course, other real values can be selected as well. Given Fredholm integral equation of the second kind b K(x, t)u(t)dt, (4.163) u(x) = f (x) + λ a

where u(x) is the unknown function to be determined, K(x, t) is the kernel, and λ is a parameter. The successive approximations method introduces the recurrence relation u0 (x) = any selective real valued function, b (4.164) K(x, t)un (t)dt, n 0. un+1 (x) = f (x) + λ a

The question of convergence of un (x) is justiﬁed by Theorem 3.1. At the limit, the solution is determined by using the limit u(x) = lim un+1 (x). n→∞

(4.165)

It is interesting to point out that the Adomian decomposition method admits the use of an iteration formula of the form u0 (x) = all terms not included inside the integral sign, b u1 (x) = λ K(x, t)u0 (t)dt, a

4.2 Fredholm Integral Equations of the Second Kind

u2 (x) = λ

147

b

K(x, t)u1 (t)dt,

(4.166)

a

.. .

b

un+1 (x) = un (x) + λ

K(x, t)un (t)dt. a

The diﬀerence between the two formulas (4.164) and (4.166) can be summarized as follows: 1. The successive approximations method gives successive approximations of the solution u(x), whereas the Adomian method gives successive components of the solution u(x). 2. The successive approximations method admits the use of a selective real-valued function for the zeroth approximation u0 , whereas the Adomian decomposition method assigns all terms that are not inside the integral sign for the zeroth component u0 (x). Recall that this assignment was modiﬁed when using the modiﬁed decomposition method. 3. The successive approximations method gives the exact solution, if it exists, by u(x) = lim un+1 (x). (4.167) n→∞

However, the Adomian decomposition method gives the solution as inﬁnite series of components by ∞ u(x) = un (x). (4.168) n=0

This series solution converges rapidly to the exact solution if such a solution exists. The successive approximations method, or the iteration method will be illustrated by studying the following examples. Example 4.23 Solve the Fredholm integral equation by using the successive approximations method u(x) = x + ex −

1

xtu(t)dt.

(4.169)

0

For the zeroth approximation u0 (x), we can select u0 (x) = 0.

(4.170)

The method of successive approximations admits the use of the iteration formula 1 x un+1 (x) = x + e − xtun (t)dt, n 0. (4.171) 0

Substituting (4.170) into (4.171) we obtain

148

4 Fredholm Integral Equations

u1 (x) = x + ex − u2 (x) = x + ex − u3 (x) = x + ex − .. .

1 0 1 0 1 0

un+1 = x + ex −

1 0

xtu0 (t)dt = ex + x, 1 xtu1 (t)dt = ex − x, 3 1 xtu2 (t)dt = ex + x, 9

xtun (t)dt = ex +

(4.172)

(−1)n x. 3n

Consequently, the solution u(x) of (4.169) is given by u(x) = lim un+1 (x) = ex . n→∞

(4.173)

Example 4.24 Solve the Fredholm integral equation by using the successive approximations method 1

xtu(t)dt.

u(x) = x + λ

(4.174)

−1

For the zeroth approximation u0 (x), we can select (4.175) u0 (x) = x. The method of successive approximations admits the use of the iteration formula 1

un+1 (x) = x + λ

−1

xtun (t)dt, n 0.

(4.176)

Substituting (4.175) into (4.176) we obtain 2 u1 (x) = x + λx, 3 2 2 2 u2 (x) = x + λx + λ2 x, 3 3 3 2 2 2 2 u3 (x) = x + λx + λ2 x + λ3 x, (4.177) 3 3 3 .. . 2 3 n+1 2 2 2 2 2 3 un+1 (x) = x + λx + λ x+ λ x + ···+ λn+1 x. 3 3 3 3 The solution u(x) of (4.174) is given by 3 3x , 0<λ< , (4.178) u(x) = lim un+1 (x) = n→∞ 3 − 2λ 2

4.2 Fredholm Integral Equations of the Second Kind

149

obtained upon using the inﬁnite geometric series for the right side of (4.177). Example 4.25 Solve the Fredholm integral equation by using the successive approximations method π 2

u(x) = sin x + sin x

cos tu(t)dt.

(4.179)

0

For the zeroth approximation u0 (x), we select u0 (x) = 0. We next use the iteration formula un+1 (x) = sin x + sin x

π 2

0

(4.180)

cos tun (t)dt,

n 0.

(4.181)

Substituting (4.180) into (4.181) we obtain 3 sin x, 2 15 u4 (x) = sin x, 8

(4.182)

u(x) = lim un+1 (x) = 2 sin x.

(4.183)

u1 (x) = sin x,

u2 (x) =

7 u3 (x) = sin x, 4 .. . 2n+1 − 1 1 sin x = 2 − un+1(x) = sin x. 2n 2n The solution u(x) of (4.179) is given by n→∞

Example 4.26 Solve the Fredholm integral equation by using the successive approximations method π u(x) = x + sec2 x −

4

xu(t)dt.

(4.184)

0

For the zeroth approximation u0 (x), we may select u0 (x) = 0. We next use the iteration formula un+1(x) = x + sec2 x −

π 4

0

xun (t)dt,

(4.185) n 0.

(4.186)

This in turn gives u1 (x) = sec2 x + x, π4 u3 (x) = sec2 x + x, 1024 .. .

π2 x, 32 π6 u4 (x) = sec2 x − x, 32768 u2 (x) = sec2 x −

(4.187)

150

4 Fredholm Integral Equations

un+1 (x) = sec2 x + (−1)n Notice that

lim

n→∞

π2 32

π2 32

n x.

n = 0.

(4.188)

Consequently, the solution u(x) of (4.184) is given by u(x) = lim un+1 (x) = sec2 x.

(4.189)

n→∞

Exercises 4.2.6 Use the successive approximations method to solve the following Fredholm integral equations: 1 1 1. u(x) = x + λ xtu(t)dt 2. u(x) = 1 + x3 + λ xtu(t)dt −1

3. u(x) = ex + 2xe−1 − 5. u(x) = 2x + sec2 x −

−1

1 −1

π 4

−π

4. u(x) = 2 + 2x + ex −

xtu(t)dt

1 0

xtu(t)dt

xu(t)dt

4

π π 4 xu(t)dt 6. u(x) = 1 + 1 + x + sec2 x − 4 0 π π 2 xtu(t)dt 7. u(x) = − 1 x + cos x − 2 0 3 π π xtu(t)dt 8. u(x) = + 3 x − cos x − 3 0 π π π3 2 2 xtu(t)dt 10. u(x) = x + sin x − xtu(t)dt 9. u(x) = x − sin x − 24 0 0 π π 3 11. u(x) = 1 + x(1 + u(t))dt x + sec x tan x − 3 0 π 3 3 x sec tu(t)dt 12. u(x) = x + sec x tan x − 2 0 π √ 4 13. u(x) = 2 x + sec x + tan x − x sec tu(t)dt π 14. u(x) = x + sin x + cos x − 8

0

0

π 4

x sin tu(t)dt

15. u(x) = (π + 2)x + sin x − cos x − 16. u(x) = x + ln(xt) −

π

0

1 0+

x(3 + u(t))dt

x(1 + u(t))dt

4.2 Fredholm Integral Equations of the Second Kind

151

4.2.7 The Series Solution Method A real function u(x) is called analytic if it has derivatives of all orders such that the Taylor series at any point b in its domain k u(n) (b) (4.190) u(x) = (x − b)n , n! n=0 converges to f (x) in a neighborhood of b. For simplicity, the generic form of Taylor series at x = 0 can be written as ∞ u(x) = an xn . (4.191) n=0

Following the discussion presented before in Chapter 3, the series solution method that stems mainly from the Taylor series for analytic functions, will be used for solving Fredholm integral equations. We will assume that the solution u(x) of the Fredholm integral equations b K(x, t)u(t)dt (4.192) u(x) = f (x) + λ a

is analytic, and therefore possesses a Taylor series of the form given in (4.191), where the coeﬃcients an will be determined recurrently. Substituting (4.191) into both sides of (4.192) gives ∞ b ∞ n n dt, (4.193) an x = T (f (x)) + λ K(x, t) an t n=0

a

or for simplicity we use a0 + a1 x + a2 x2 + · · · = T (f (x)) + λ

n=0

b

K(x, t) a0 + a1 t + a2 t2 + · · · dt,

a

(4.194) where T (f (x)) is the Taylor series for f (x). The integral equation (4.192) will be converted to a traditional integral in (4.193) or (4.194) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integral in (4.193) or (4.194), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x in both sides of the resulting equation to obtain a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (4.191). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained

152

4 Fredholm Integral Equations

series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. It is worth noting that using the series solution method for solving Fredholm integral equations gives exact solutions if the solution u(x) is a polynomial. However, if the solution is any other elementary function such as sin x, ex , etc, the series method gives the exact solution after rounding few of the coeﬃcients aj , j 0. This will be illustrated by studying the following examples. Example 4.27 Solve the Fredholm integral equation by using the series solution method 1 u(x) = (x + 1)2 + (xt + x2 t2 )u(t)dt. (4.195) −1

Substituting u(x) by the series u(x) =

∞

a n xn

(4.196)

n=0

into both sides of Eq. (4.195) leads to 1 ∞ ∞ n 2 2 2 n an x = (x + 1) + (an t ) dt. (xt + x t ) −1

n=0

(4.197)

n=0

Evaluating the integral at the right side gives 2 2 2 2 a0 + a1 x + a2 x2 + a3 x3 + · · · = 1 + 2 + a1 + a3 + a5 + a7 x 3 5 7 9 2 2 2 2 2 + 1 + a0 + a2 + a4 + a6 + a8 x2 . 3 5 7 9 11 (4.198) Equating the coeﬃcients of like powers of x in both sides of (4.198) gives 25 , an = 0, n 3. a0 = 1, a1 = 6, a2 = (4.199) 9 The exact solution is given by 25 (4.200) u(x) = 1 + 6x + x2 , 9 obtained upon substituting (4.199) into (4.196). Example 4.28 Solve the Fredholm integral equation by using the series solution method 1 2 3 (1 + xt)u(t)dt. (4.201) u(x) = x − x + 0

Substituting u(x) by the series u(x) =

∞ n=0

an xn ,

(4.202)

4.2 Fredholm Integral Equations of the Second Kind

into both sides of Eq. (4.201) leads to 1 ∞ ∞ n 2 3 n an x = x − x + (an t ) dt. (1 + xt) 0

n=0

153

(4.203)

n=0

Evaluating the integral at the right side, and equating the coeﬃcients of like powers of x in both sides of the resulting equation we ﬁnd 29 1 a0 = − , a1 = − , a2 = 1, a3 = −1, an = 0, n 4. (4.204) 60 6 Consequently, the exact solution is given by 29 1 u(x) = − − x + x2 − x3 . (4.205) 60 6 Example 4.29 Solve the Fredholm integral equation by using the series solution method 1 (xt2 − x2 t)u(t)dt. (4.206) u(x) = −x4 + −1

Substituting u(x) by the series u(x) =

∞

an xn ,

(4.207)

n=0

into both sides of Eq. (4.206) leads to 1 ∞ ∞ n 4 2 2 n (xt − x t) an x = −x + an t dt. n=0

−1

(4.208)

n=0

Evaluating the integral at the right side, and equating the coeﬃcients of like powers of x in both sides of the resulting equation we ﬁnd 30 20 a0 = 0, a1 = − , a2 = , a3 = 0, a4 = −1 (4.209) 133 133 an = 0, n 5. Consequently, the exact solution is given by 20 2 30 x+ x − x4 . (4.210) u(x) = − 133 133 Example 4.30 Solve the Fredholm integral equation by using the series solution method π2 u(x) = −1 + cos x + u(t)dt. (4.211) 0

Substituting u(x) by the series u(x) =

∞ n=0

into both sides of Eq. (4.211) gives

an xn ,

(4.212)

154

4 Fredholm Integral Equations ∞

an xn = −1 + cos x +

n=0

π 2

0

∞

(an tn ) dt.

(4.213)

n=0

Evaluating the integral at the right side, using the Taylor series of cos x, and proceeding as before we ﬁnd (−1)j , j 0. (4.214) a0 = 1, a2j+1 = 0, a2j = (2j)! Consequently, the exact solution is given by u(x) = cos x. (4.215)

Exercises 4.2.7 Use the series solution method to solve the following Fredholm integral equations: 1 1 (1 − 3xt)u(t)dt 2. u(x) = 6x + 4x2 + (xt2 − x2 t)u(t)dt 1. u(x) = 1 + 0

3. u(x) = 5x − 2x2 + 4. u(x) = 5x +

1

−1

−1

1

−1

(x2 t3 − x3 t2 )u(t)dt

(1 − xt)u(t)dt

5. u(x) = 2 + 5x − 3x2 + 6. u(x) = 3x − 5x3 +

1

−1

1

−1

(1 − xt)u(t)dt

(1 − xt)u(t)dt

7. u(x) = 2 − 2x + 5x4 + 7x5 + 8. u(x) = 3x2 − 5x3 +

1 −1

9. u(x) = −2 + sin x + 11. u(x) = sec2 x − 1 +

π 2

−π

1 −1

(x − t)u(t)dt

x4 tu(t)dt tu(t)dt

10. u(x) = −2 + x2 + sin x +

2

π 4

0

u(t)dt

12. u(x) = −1 + ln(1 + x) +

π 2

−π

tu(t)dt

2

e−1

0

u(t)dt

4.3 Homogeneous Fredholm Integral Equation Substituting f (x) = 0 into the Fredholm integral equation of the second kind b u(x) = f (x) + λ K(x, t)u(t)dt, (4.216) a

the homogeneous Fredholm integral equation of the second kind is given by

4.3 Homogeneous Fredholm Integral Equation

155

b

u(x) = λ

K(x, t)u(t)dt.

(4.217)

a

In this section we will focus our study on the homogeneous Fredholm integral equation (4.217) for separable kernel K(x, t) only. The main goal for studying the homogeneous Fredholm equation is to ﬁnd nontrivial solution, because the trivial solution u(x) = 0 is a solution of this equation. Moreover, the Adomian decomposition method is not applicable here because it depends mainly on assigning a non-zero value for the zeroth component u0 (x), and in this kind of equations f (x) = 0. Based on this, the direct computation method will be employed here to handle this kind of equations.

4.3.1 The Direct Computation Method The direct computation method was used before in this chapter. This method replaces the homogeneous Fredholm integral equations by a single algebraic equation or by a system of simultaneous algebraic equations depending on the number of terms of the separable kernel K(x, t). As stated before, the direct computation method handles Fredholm integral equations in a direct manner and gives the solution in an exact form but not in a series form as Adomian method or the successive approximations method. It is important to point out that this method will be applied for the degenerate or separable kernels of the form n gk (x)hk (t). (4.218) K(x, t) = k=1

The direct computation method can be applied as follows: 1. We ﬁrst substitute (4.218) into the homogeneous Fredholm integral equation the form: b u(x) = λ K(x, t) u(t)dt. (4.219) a

2. This substitution leads to b h1 (t)u(t)dt + λg2 (x) u(x) = λg1 (x) a

b

h2 (t)u(t)dt + · · · a

b

+λgn (x)

(4.220)

hn (t)u(t)dt. a

3. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (4.220) becomes u(x) = λα1 g1 (x) + λα2 g2 (x) + · · · + λαn gn (x), where

(4.221)

156

4 Fredholm Integral Equations

b

hi (t) u(t)dt, 1 i n.

αi =

(4.222)

a

4. Substituting (4.221) into (4.222) gives a system of n simultaneous algebraic equations that can be solved to determine the constants αi , 1 i n. Using the obtained numerical values of αi into (4.221), the solution u(x) of the homogeneous Fredholm integral equation (4.217) follows immediately. Example 4.31 Solve the homogeneous Fredholm integral equation by using the direct computation method π2 cos x sin t u(t)dt. (4.223) u(x) = λ 0

This equation can be rewritten as u(x) = αλ cos x, where

π 2

α=

sin t u(t)dt.

(4.224) (4.225)

0

Substituting (4.224) into (4.225) gives π2 cos t sin tdt, α = αλ

(4.226)

0

that gives 1 αλ. (4.227) 2 Recall that α = 0 gives the trivial solution. For α = 0, we ﬁnd that the eigenvalue λ is given by λ = 2. (4.228) α=

This in turn gives the eigenfunction u(x) by u(x) = A cos x,

(4.229)

where A is a non zero arbitrary constant, with A = 2α. Example 4.32 Solve the homogeneous Fredholm integral equation by using the direct computation method 1 2ex+t u(t)dt. (4.230) u(x) = λ 0

This equation can be rewritten as u(x) = 2αλex , where

α= 0

1

et u(t)dt.

(4.231) (4.232)

4.3 Homogeneous Fredholm Integral Equation

Substituting (4.231) into (4.232) gives α = 2αλ

1

157

e2t dt,

(4.233)

0

that gives

α = αλ(e2 − 1).

(4.234)

Recall that α = 0 gives the trivial solution. For α = 0, we ﬁnd that the eigenvalue λ is given by 1 . (4.235) λ= 2 e −1 This in turn gives the eigenfunction u(x) by A u(x) = 2 ex , e −1 where A is a non zero arbitrary constant, with A = 2α.

(4.236)

Example 4.33 Solve the homogeneous Fredholm integral equation by using the direct computation method π sin(x + t) u(t)dt. (4.237) u(x) = λ 0

Notice that the kernel sin(x+t) = sin x cos t+cos x sin t is separable. Equation (4.237) can be rewritten as u(x) = αλ sin x + βλ cos x,

where

π

cos tu(t)dt,

α= 0

β=

(4.238)

π

sin tu(t)dt.

(4.239)

0

Substituting (4.238) into (4.239) gives π α= cos t(αλ sin t + βλ cos t)dt, 0

(4.240)

π

sin t(αλ sin t + βλ cos t)dt,

β= 0

that gives 1 1 βλπ, β = αλπ. 2 2 For α = 0, β = 0, we ﬁnd that the eigenvalue λ is given by 2 λ = ± , α = β. π This in turn gives the eigenfunction u(x) by A u(x) = ± (sin x + cos x), π α=

(4.241)

(4.242)

(4.243)

158

4 Fredholm Integral Equations

where A = 2α. Example 4.34 Solve the homogeneous Fredholm integral equation by using the direct computation method 1 u(x) = λ (12x + 2t) u(t)dt. (4.244) −1

Equation (4.244) can be rewritten as u(x) = 12αλx + 2βλ, where

1

u(t)dt,

α=

1

β=

−1

(4.245)

tu(t)dt.

(4.246)

−1

Substituting (4.245) into (4.246) gives 1 (12αλt + 2βλ)dt = 4βλ, α= −1 1

β=

(4.247) t(12αλt + 2βλ)dt = 8αλ.

−1

Recall that α = 0 and β = 0 give the trivial solution. For α = 0, β = 0, we ﬁnd that the eigenvalue λ is given by √ 1 λ = ± √ , β = 2α. (4.248) 4 2 This in turn gives the eigenfunction u(x) by √ α (4.249) u(x) = ± √ (6x + 2). 2 2 Example 4.35 Solve the homogeneous Fredholm integral equation by using the direct computation method 1 u(x) = λ 10(x2 − 2xt − t2 ) u(t)dt. (4.250) 0

Equation (4.250) can be rewritten as u(x) = 10αλx2 − 20βλx − 10γλ, where

1

u(t)dt,

α= 0

β=

1

tu(t)dt, 0

γ=

1

t2 u(t)dt.

(4.251)

(4.252)

0

Substituting (4.251) into (4.252) gives 1 10 α= αλ − 10βλ − 10λγ, (10αλt2 − 20βλt − 10γλ)dt = 3 0 1 20 5 β= t(10αλt2 − 20βλt − 10γλ)dt = αλ − βλ − 5λγ, (4.253) 2 3 0

4.4 Fredholm Integral Equations of the First Kind

1

γ= 0

159

t2 (10αλt2 − 20βλt − 10γλ)dt = 2αλ − 5βλ −

10 λγ. 3

Recall that α = 0, β = 0 and γ = 0 give the trivial solution. For α = 0, β = 0 and γ = 0, and by solving the system of equations we ﬁnd 20 7 3 γ, β = γ, (4.254) λ=− , α= 5 3 3 and γ is left as a free parameter. This in turn gives the eigenfunction u(x) by (4.255) u(x) = γ(−40x2 + 28x + 6), where γ is a non-zero arbitrary constant.

Exercises 4.3 Use the direct computation method to solve the homogeneous Fredholm integral equations: π π 3 1. u(x) = λ sin2 xu(t)dt 2. u(x) = λ tan x sec tu(t)dt 3. u(x) = λ 5. u(x) = λ 7. u(x) = λ 9. u(x) = λ

0

π 4

10 sec2 xu(t)dt

4. u(x) = λ

xtu(t)dt

6. u(x) = λ

8 sin−1 xtu(t)dt

8. u(x) = λ

(x + t)u(t)dt

10. u(x) = λ

0

2 0 1 0

−1

0

π

sin xu(t)dt

0 1

π

1

0 1 −1

12. u(x) = λ

(2 − 3x − 3t)u(t)dt

14. u(x) = λ

8 cos −1 xtu(t)dt

0

1 cos(x − t)u(t)dt π

xet u(t)dt

0

1

11. u(x) = λ 13. u(x) = λ

1

−1 1 0 1

(x − 10t2 )u(t)dt

(3 − 6x + 9t)u(t)dt (−12x2 + 24xt + 18t2 )u(t)dt

0

4.4 Fredholm Integral Equations of the First Kind We close this chapter by studying Fredholm integral equations of the ﬁrst kind given by b f (x) = λ K(x, t)u(t)dt, x ∈ D, (4.256) a

where D is a closed and bounded set in real numbers, and f (x) is the data. The range of x does not necessarily coincide with the range of integration [7]. The unknown function u(x), that will be determined, occurs only inside the integral sign and this causes special diﬃculties. The kernel K(x, t) and the function f (x) are given real-valued functions, and λ is a parameter that

160

4 Fredholm Integral Equations

is often omitted. However, the parameter λ plays an important role in the singular cases and in the bifurcation points as will be seen later in the text. An important remark has been reported in [7] and other references concerning the data function f (x). The function f (x) must lie in the range of the kernel K(x, t) [7]. For example, if we set the kernel by K(x, t) = sin x sin t. (4.257) Then if we substitute any integrable function u(x) in (4.256), and we evaluate the integral, the resulting f (x) must clearly be a multiple of sin x [7]. This means that if f (x) is not a multiple of the x component of the kernel, then a solution for (4.256) does not exist. This necessary condition on f (x) can be generalized. In other words, the data function f (x) must contain components which are matched by the corresponding x components of the kernel K(x, t) Fredholm integral equation of the ﬁrst kind is considered ill-posed problem. Hadamard [8] postulated the following three properties: 1. Existence of a solution. 2. Uniqueness of a solution. 3. Continuous dependence of the solution u(x) on the data f (x). This property means that small errors in the data f (x) should cause small errors [9] in the solution u(x). A problem is called a well-posed problem if it satisﬁes the three aforementioned properties. Problems that are not well-posed are called ill-posed problems such as inverse problems. Inverse problems are ill-posed problems that might not have a solution in the true sense, if a solution exists it may not be unique, and the obtained solution might not depend continuously on the observed data. If the kernel K(x, t) is smooth, then the Fredholm integral equation (4.256) is very often ill-posed and the solution u(x) is very sensitive to any change in the data f (x). In other words, a very small change on the data f (x) can give a large change in the solution u(x). For all these reasons, the Fredholm integral equations of the ﬁrst kind is ill-posed that may have no solution, or if a solution exists it is not unique and may not depend continuously on the data. The Fredholm integral equations of the ﬁrst kind (4.256) appear in many physical models such as radiography, stereology, spectroscopy, cosmic radiation, image processing and electromagnetic ﬁelds. Fredholm integral equations of the ﬁrst kind arise naturally in the theory of signal processing. Many inverse problems in science and engineering lead to the Fredholm integral equations of the ﬁrst kind. An inverse problem is a process where the solution u(x) can be obtained by solving (4.256) from the observed data f (x) at various values of x. Most inverse problems are ill-posed problems. This means that the Fredholm integral equations of the ﬁrst kind is aill-posed problem, and solving this equation may lead to a lot of diﬃculties. Several methods have been used to handle the Fredholm integral equations of the ﬁrst kind. The Legendre wavelets, the augmented Galerkin method, and the collocation method are examples of the methods used to handle this

4.4 Fredholm Integral Equations of the First Kind

161

equation. The methods that we used so far in this text cannot handle this kind of equations independently if it is expressed in its standard form (4.256). However, in this text, we will ﬁrst apply the method of regularization that received a considerable amount of interest, especially in solving ﬁrst order integral equations. The method transforms ﬁrst kind equation to second kind equation. We will second apply the homotopy perturbation method [10] to handle speciﬁc cases of the Fredholm integral equations where the kernel K(x, t) is separable. In what follows we will present a brief summary of the method of regularization and the homotopy perturbation method that will be used to handle the Fredholm integral equations of the ﬁrst kind.

4.4.1 The Method of Regularization The method of regularization was established independently by Phillips [11] and Tikhonov [12]. The method of regularization consists of replacing illposed problem by well-posed problem. The method of regularization transforms the linear Fredholm integral equation of the ﬁrst kind b f (x) = K(x, t)u(t)dt, x ∈ D, (4.258) a

to the approximation Fredholm integral equation b K(x, t)uμ (t)dt, x ∈ D, μuμ (x) = f (x) −

(4.259)

a

where μ is a small positive parameter. It is clear that (4.259) is a Fredholm integral equation of the second kind that can be rewritten 1 b 1 uμ (x) = f (x) − K(x, t)uμ (t)dt, x ∈ D. (4.260) μ μ a Moreover, it was proved in [7,13] that the solution uμ of equation (4.260) converges to the solution u(x) of (4.258) as μ → 0 according to the following lemma [14]: Lemma 4.1 Suppose that the integral operator of (4.258) is continuous and coercive in the Hilbert space where f (x), u(x), and uμ (x) are deﬁned, then: 1. |uμ | is bounded independently of μ, and 2. |uμ (x) − u(x)| → 0 when μ → 0. The proof of this lemma can be found in [7,13]. In summary, by combining the method of regulariztion with any of the methods used before for solving Fredholm integral equation of the second

162

4 Fredholm Integral Equations

kind, we can solve Fredholm integral equation of the ﬁrst kind (4.258). The method of regulariztion transforms the ﬁrst kind to a second kind. The resulting integral equation (4.260) can be solved by any of the methods that were presented before in this chapter. The exact solution u(x) of (4.258) can thus be obtained by u(x) = lim uμ (x). (4.261) μ→0

In what follows we will present ﬁve illustrative examples where we will use the method of regulariztion to transform the ﬁrst kind equation to a second kind equation. The resulting equation will be solved by any appropriate method that we used before. Example 4.36 Combine the method of regulariztion and the direct computation method to solve the Fredholm integral equation of the ﬁrst kind 14 1 x e = ex−t u(t)dt. (4.262) 4 0 Using the method of regularization, Equation (4.262) can be transformed to 1 1 x 1 4 x−t e uμ (t)dt. (4.263) uμ (x) = e − 4μ μ 0 The resulting Fredholm integral equation of the second kind will be solved by the direct computation method. Equation (4.263) can be written as 1 α (4.264) uμ (x) = − ex , 4μ μ where 1 4

α= 0

e−t uμ (t)dt.

(4.265)

To determine α, we substitute (4.264) into (4.265), integrate the resulting integral and solve to ﬁnd that 1 α= . (4.266) 1 + 4μ This in turn gives ex uμ (x) = . (4.267) 1 + 4μ The exact solution u(x) of (4.262) can be obtained by u(x) = lim uμ (x) = ex . μ→0

(4.268)

Example 4.37 Combine the method of regulariztion and the direct computation method to solve the Fredholm integral equation of the ﬁrst kind

4.4 Fredholm Integral Equations of the First Kind

ex + 1 =

1

163

(4tex + 3) u(t)dt.

(4.269)

0

Notice that the data function f (x) = ex + 1 contains components which are matched by the corresponding x components of the kernel K(x, t) = 4tex + 3. This is a necessary condition to guarantee a solution. Using the method of regularization, Equation (4.269) can be transformed to 1 1 1 1 (4tex + 3)uμ (t)dt. (4.270) uμ (x) = ex + − μ μ μ 0 The resulting Fredholm integral equation of the second kind will be solved by the direct computation method. Equation (4.270) can be written as 1 1 4α x 3β uμ (x) = − e + − , (4.271) μ μ μ μ where 1

α= 0

1

tuμ (t)dt,

β= 0

uμ (t)dt.

(4.272)

To determine α and β, we substitute (4.271) into (4.272), integrate the resulting integrals and solve to ﬁnd that −2(e + 6 + μe) 3(e − 3 − μ) , β=− α= . (4.273) 2(6e − 18 − 7μ − μ2 ) 6e − 18 − 7μ − μ2 Substituting this result into (4.271) gives the approximate solution (1 + μ)ex + (7 − 3e + μ) . (4.274) uμ (x) = 6(3 − e) + (7μ + μ2 ) The exact solution u(x) of (4.269) can be obtained by 1 7 − 3e u(x) = lim uμ (x) = ex + . (4.275) μ→0 6(3 − e) 6(3 − e) It is interesting to point out that another solution to this equation is given by (4.276) u(x) = x2 . As stated before, the Fredholm integral equation of the ﬁrst kind is ill-posed problem. For ill-posed problems, the solution might not exist, and if it exists, the solution may not be unique. Example 4.38 Combine the method of regulariztion and the direct computation method to solve the Fredholm integral equation of the ﬁrst kind π π sin x = sin(x − t) u(t)dt. (4.277) 2 0 Notice that the data function f (x) = π2 sin x contains component which is matched by the corresponding x component of the kernel K(x, t). This is a necessary condition to guarantee a solution.

164

to

4 Fredholm Integral Equations

Using the method of regularization, Equation (4.277) can be transformed 1 π π uμ (x) = sin x − sin(x − t)uμ (t)dt, (4.278) 2μ μ 0

that can be written as

uμ (x) =

where

α π − 2μ μ

sin x +

π

α= 0

cos tuμ (t)dt,

β= 0

β cos x, μ

(4.279)

sin tuμ (t)dt.

(4.280)

π

To determine α and β, we substitute (4.279) into (4.280), integrate the resulting integrals and solve to ﬁnd that π2 μ π3 . (4.281) , β = α= 2(π 2 + 4μ2 ) π 2 + 4μ2 Substituting this result into (4.279) gives the approximate solution 2πμ π2 uμ (x) = 2 sin x + 2 cos x. (4.282) 2 π + 4μ π + 4μ2 The exact solution u(x) of (4.277) can be obtained by (4.283) u(x) = lim uμ (x) = cos x. μ→0

Example 4.39 Combine the method of regulariztion and the Adomian decomposition method to solve the Fredholm integral equation of the ﬁrst kind 13 1 −x e = et−x u(t)dt. (4.284) 3 0 Using the method of regularization, Equation (4.284) can be transformed to 1 1 −x 1 3 t−x e uμ (t)dt. (4.285) uμ (x) = e − 3μ μ 0 The Adomian decomposition method admits the use of ∞ uμ (x) = uμn (x), (4.286) n=0

and the recurrence relation 1 −x e , 3μ 1 1 3 t−x e uμk (t)dt, uμk+1 (x) = − μ 0 This in turn gives the components uμ0 (x) =

(4.287) k 0.

4.4 Fredholm Integral Equations of the First Kind

165

1 −x 1 e , uμ1 (x) = − 2 e−x , 3μ 9μ (4.288) 1 −x 1 −x e , u (x) = − e , uμ2 (x) = μ 3 27μ3 81μ2 and so on. Substituting this result into (4.286) gives the approximate solution 1 (4.289) uμ (x) = e−x . 1 + 3μ The exact solution u(x) of (4.284) can be obtained by uμ0 (x) =

u(x) = lim uμ (x) = e−x . μ→0

(4.290)

Example 4.40 Combine the method of regulariztion and the successive approximations method to solve the Fredholm integral equation of the ﬁrst kind 1 1 xt u(t)dt. (4.291) x= 4 0 Using the method of regularization, Equation (4.291) can be transformed to 1 1 1 xtuμ (t)dt. (4.292) uμ (x) = x− 4μ μ 0 To use the successive approximations method, we ﬁrst select uμ0 (x) = 0. Consequently, we obtain the following approximations uμ0 (x) = 0, 1 x, uμ1 (x) = 4μ 1 1 uμ2 (x) = x, x− (4.293) 4μ 12μ2 1 1 1 x+ x, uμ3 (x) = x− 4μ 12μ2 36μ3 1 1 1 1 x+ x− x, uμ4 (x) = x− 4μ 12μ2 36μ3 108μ4 and so on. Based on this we obtain the approximate solution 3 x. (4.294) uμ (x) = 4(1 + 3μ) The exact solution u(x) of (4.291) can be obtained by 3 u(x) = lim uμ (x) = x. (4.295) μ→0 4 It is interesting to point out that another solution to this equation is given by (4.296) u(x) = x2 . As stated before, the Fredholm integral equation of the ﬁrst kind is ill-posed problem and the solution may not be unique.

166

4 Fredholm Integral Equations

Exercises 4.4.1 Combine the regularization method with any other method to solve the Fredholm integral equations of the ﬁrst kind 1 1 1 1 2 1. (1 − e−2 )e3x = e3x−4tu(t)dt 2. e3x = e3x−3t u(t)dt 2 2 0 0 1 1 6 3 xt2 u(t)dt 4. x2 = x2 t2 u(t)dt 3. x = 4 5 0 0 1 1 2 1 5. x2 = x2 t2 u(t)dt 6. x = xtu(t)dt 5 5 −1 0 1 1 2 1 x2 t2 u(t)dt 8. x2 = x2 t2 u(t)dt 7. x2 = 6 3 0 −1 1 1 1 1 9. − x = xtu(t)dt 10. x = xtu(t)dt 4 4 0 0 1 1 1 7 11. x= x= xtu(t)dt 12. xtu(t)dt 12 12 0 0 π π π π 13. sin x = cos(x − t)u(t)dt 14. cos x = cos(x − t)u(t)dt 2 2 0 0 π π (x − t)u(t)dt 16. 2 + π − 2x = (x − t)u(t)dt 15. 2 − π + 2x = 0

0

4.4.2 The Homotopy Perturbation Method The homotopy perturbation method was introduced and developed by JiHuan He in [10] and was used recently in the literature for solving linear and nonlinear problems. The homotopy perturbation method couples a homotopy technique of topology and a perturbation technique. A homotopy with an embedding parameter p ∈ [0, 1] is constructed, and the impeding parameter p is considered a small parameter. The method was derived and illustrated in [10], and several diﬀerential equations were examined. The coupling of the perturbation method and the homotopy method has eliminated the limitations of the traditional perturbation technique [10]. In what follows we illustrate the homotopy perturbation method to handle Fredholm integral equations of the second kind and the ﬁrst kind.

The HPM for Fredholm Integral Equation of the Second Kind In what follows we present the homotopy perturbation method for handling the Fredholm integral equations of the second kind. We ﬁrst consider the Fredholm integral equation of the second kind

4.4 Fredholm Integral Equations of the First Kind

167

b

v(x) = f (x) +

K(x, t)v(t)dt.

(4.297)

a

We now deﬁne the operator

b

L(u) = u(x) − f (x) −

K(x, t)u(t)dt = 0,

(4.298)

a

where u(x) = v(x). Next we deﬁne the homotopy H(u, p), p ∈ [0, 1] by H(u, 0) = F (u), H(u, 1) = L(u), (4.299) where F (u) is a functional operator. We construct a convex homotopy of the form H(u, p) = (1 − p)F (u) + pL(u) = 0. (4.300) This homotopy satisﬁes (4.299) for p = 0 and p = 1 respectively. The embedding parameter p monotonically increases from 0 to 1 as the trivial problem F (u) = 0 continuously deformed [10] to the original problem L(u) = 0. The homotopy perturbation method admits the use of the expansion ∞ u= pn un , (4.301) n=0

and consequently v = lim

p→1

∞

pn un .

(4.302)

n=0

The series (4.302) converges to the exact solution if such a solution exists. Substituting (4.301) into (4.300), using F (x) = u(x) − f (x), and equating the terms with like powers of the embedding parameter p we obtain the recurrence relation b 0 n+1 p : u0 (x) = f (x), p : un+1 = K(x, t)un (t)dt, n 0. (4.303) a

Notice that the recurrence relation (4.303) is the same standard Adomian decomposition method as presented before in this chapter. This proves the following theorem: Theorem 4.3 The Adomian decomposition method is a homotopy perturbation method with a convex homotopy given by b K(x, t)un (t)dt = 0. (4.304) H(u, p) = u(x) − f (x) − p a

HPM for Fredholm Integral Equation of the First Kind In what follows we present the homotopy perturbation method for handling the Fredholm integral equations of the ﬁrst kind of the form b f (x) = K(x, t)v(t)dt. (4.305) a

168

4 Fredholm Integral Equations

We now deﬁne the operator

L(u) = f (x) −

b

K(x, t)u(t)dt = 0.

(4.306)

a

We construct a convex homotopy of the form H(u, p) = (1 − p)u(x) + pL(u)(x) = 0. (4.307) The embedding parameter p monotonically increases from 0 to 1. The homotopy perturbation method admits the use of the expansion ∞ p n un , (4.308) u= n=0

and consequently v(x) = lim

p→1

∞

pn un (x).

(4.309)

n=0

The series (4.309) converges to the exact solution if such a solution exists. Substituting (4.308) into (4.307), and proceeding as before we obtain the recurrence relation u0 (x) = 0, u1 (x) = f (x), b (4.310) un+1 (x) = un (x) − K(x, t)un (t)dt, n 1. a

If the kernel is separable, i.e. K(x, t) = g(x)h(t), then the following condition b K(t, t)dt < 1, (4.311) 1 − a must be justiﬁed for convergence. The proof of this condition is left to the reader. We will concern ourselves only on the case where K(x, t) = g(x)h(t). The HPM will be used to solve the following Fredholm integral equations of the ﬁrst kind. Example 4.41 Use the homotopy perturbation method to solve the Fredholm integral equation of the ﬁrst kind 13 1 x e = ex−t u(t)dt. (4.312) 3 0 Notice that

13 2 K(t, t)dt = < 1. 1 − 3 0

Using the recurrence relation (4.310) we ﬁnd

(4.313)

4.4 Fredholm Integral Equations of the First Kind

u0 (x) = 0,

1 x e , 3

u1 (x) =

un+1 (x) = un (x) −

1 3

169

e

x−t

0

un (t)dt,

(4.314) n 1.

This in turn gives u0 (x) = 0,

u1 (x) =

u2 (x) = u1 (x) −

1 3

0

u3 (x) = u2 (x) −

1 3

0

1 3

1 x e , 3 ex−tu1 (t)dt = e

x−t

2 x e , 9

4 x e , u2 (t)dt = 27

8 x e , 81 0 and so on. Consequently, the approximate solution is given by 1 2 4 8 x u(x) = e + + + + ··· , 3 9 27 81 that converges to the exact solution u(x) = ex . u4 (x) = u3 (x) −

(4.315)

ex−tu3 (t)dt =

(4.316)

(4.317)

Example 4.42 Use the homotopy perturbation method to solve the Fredholm integral equation of the ﬁrst kind 14 1 −x e = et−x u(t)dt. (4.318) 4 0 14 3 K(t, t)dt = < 1. 1 − 4 0

Notice that

Using the recurrence relation (4.310) we ﬁnd 1 u0 (x) = 0, u1 (x) = e−x , 4 14 un+1 (x) = un (x) − et−x un (t)dt, 0

(4.319)

(4.320) n 1.

This in turn gives u0 (x) = 0,

u1 (x) =

u2 (x) = u1 (x) −

0

u3 (x) = u2 (x) −

1 4

0

1 4

1 −x e , 4 et−x u1 (t)dt =

3 −x e , 16

et−x u2 (t)dt =

9 −x e , 64

(4.321)

170

4 Fredholm Integral Equations

u4 (x) = u3 (x) −

1 4

0

et−x u3 (t)dt =

27 −x e , 256

and so on. Consequently, the approximate solution is given by 1 3 9 27 u(x) = e−x + + + + ··· , 4 16 64 256 that converges to the exact solution u(x) = e−x ,

(4.322)

(4.323)

obtained by evaluating the sum of the inﬁnite geometric series. Example 4.43 Use the homotopy perturbation method to solve the Fredholm integral equation of the ﬁrst kind 1

xt u(t)dt.

(4.324)

2 K(t, t)dt = < 1. 3

(4.325)

x= 0

Notice that

1 −

0

1

Using the recurrence relation (4.310) we ﬁnd u0 (x) = 0, u1 (x) = x, 1 un+1(x) = un (x) − xtun (t)dt, 0

n 1.

(4.326)

This in turn gives u0 (x) = 0,

u1 (x) = x, 1 2 xtu1 (t)dt = x, u2 (x) = u1 (x) − 3 0 1 4 xtu2 (t)dt = x, u3 (x) = u2 (x) − 9 0 1 8 xtu3 (t)dt = u4 (x) = u3 (x) − x, 27 0 and so on. Consequently, the approximate solution is given by 8 2 4 + ··· , u(x) = x 1 + + + 3 9 27 that converges to the exact solution u(x) = 3x.

(4.327)

(4.328)

(4.329)

Example 4.44 Use the homotopy perturbation method to solve the Fredholm integral equation of the ﬁrst kind 1 5 x= xt u(t)dt. (4.330) 6 0

4.4 Fredholm Integral Equations of the First Kind

Notice that

1 −

1

0

171

2 K(t, t)dt = < 1. 3

Using the recurrence relation (4.310) we ﬁnd 5 u0 (x) = 0, u1 (x) = x, 6 1 un+1(x) = un (x) − xtun (t)dt, 0

(4.331)

(4.332) n 1.

This in turn gives u0 (x) = 0,

u1 (x) =

u2 (x) = u1 (x) − u3 (x) = u2 (x) −

1

xtu1 (t)dt =

0

1

0

5 x, 6 5 x, 9

10 x, xtu2 (t)dt = 27

(4.333)

1

20 x, 81 0 and so on. Consequently, the approximate solution is given by 5 2 4 8 5 + ··· , (4.334) u(x) = x + x 1 + + + 6 9 3 9 27 that converges to the exact solution 5 u(x) = x, (4.335) 2 obtained by evaluating the sum of the inﬁnite geometric series. It is interesting to point out that another solution to this equation is given by u(x) = 1 + x. (4.336) u4 (x) = u3 (x) −

xtu3 (t)dt =

As stated before, the Fredholm integral equation of the ﬁrst kind is ill-posed problem. For ill-posed problems, the solution might not exist, and if it exists, the solution may not be unique. Example 4.45 Use the homotopy perturbation method to solve the Fredholm integral equation of the ﬁrst kind 1 1 x= xt u(t)dt. (4.337) 4 0 Notice that

1 −

0

1

2 K(t, t)dt = < 1. 3

Using the recurrence relation (4.310) we ﬁnd

(4.338)

172

4 Fredholm Integral Equations

1 u1 (x) = x, 4 1 un+1(x) = un (x) − xtun (t)dt, u0 (x) = 0,

0

(4.339) n 1.

This in turn gives 1 u1 (x) = x, 4 1 1 u2 (x) = u1 (x) − xtu1 (t)dt = x, 6 0 1 1 xtu2 (t)dt = x, u3 (x) = u2 (x) − 9 0 1 2 u4 (x) = u3 (x) − x, xtu3 (t)dt = 27 0 and so on. Consequently, the approximate solution is given by 1 2 4 8 u(x) = x 1 + + + + ··· , 4 3 9 27 that converges to the exact solution 3 u(x) = x. 4 It is interesting to point out that another solution to this equation by u(x) = x2 . u0 (x) = 0,

(4.340)

(4.341)

(4.342) is given (4.343)

As stated before, the Fredholm integral equation of the ﬁrst kind is ill-posed problem and the solution may not be unique. Notice that the ill-posed Fredholm problem is linear.

Exercises 4.4.2 Use the homotopy perturbation method to solve the Fredholm integral equations of the ﬁrst kind 1 1 1 1 2 1. (1 − e−2 )e3x = e3x−4tu(t)dt 2. e3x = e3x−3tu(t)dt 2 2 0 0 1 1 6 3 xt2 u(t)dt 4. x2 = x2 t2 u(t)dt 3. x = 4 5 0 0 1 1 1 2 x2 t2 u(t)dt 6. x = xtu(t)dt 5. x2 = 5 5 −1 0 1 1 1 2 7. x2 = x2 t2 u(t)dt 8. x2 = x2 t2 u(t)dt 6 3 0 −1 1 1 1 1 9. − x = xtu(t)dt 10. x = xtu(t)dt 4 4 0 0

References 11.

1 x= 12

173

1

0

xtu(t)dt

12.

7 x= 12

1 0

xtu(t)dt

References 1. W.V. Lovitt, Linear Integral Equations, Dover, New York, (1950). 2. G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994). 3. G. Adomian, A review of the decomposition method and some recent results for nonlinear equation, Math. Comput. Modelling, 13(1992) 17–43. 4. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 5. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc Singapore, (1997). 6. A.M.Wazwaz, Necessary conditions for the appearance of noise terms in decomposition solution series, Appl. Math. Comput., 81 (1997) 199–204. 7. L.M. Delves, and J. Walsh, Numerical Solution of Integral Equations, Oxford University Press, London, (1974). 8. J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Diﬀerential equations, Yale University Press, New Haven, (1923). 9. R. Kress, Linear Integral Equations, Springer, Berlin, (1999). 10. J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg., 178 (1999) 257–262. 11. D.L. Phillips, A technique for the numerical solution of certain integral equations of the ﬁrst kind, J. Assoc. Comput. Mach, 9 (1962) 84–96. 12. A.N. Tikhonov, On the solution of incorrectly posed problem and the method of regularization, Soviet Math, 4 (1963) 1035–1038. 13. R.F. Churchhouse, Handbook of Applicable Mathematics, Wiley, New York, (1981). 14. Y. Cherruault and V. Seng, The resolution of non-linear integral equations of the ﬁrst kind using the decomposition method of Adomian, Kybernetes, 26 (1997) 109–206.

Chapter 5

Volterra Integro-Diﬀerential Equations

5.1 Introduction Volterra studied the hereditary inﬂuences when he was examining a population growth model. The research work resulted in a speciﬁc topic, where both diﬀerential and integral operators appeared together in the same equation. This new type of equations was termed as Volterra integro-diﬀerential equations [1–4], given in the form x u(n) (x) = f (x) + λ K(x, t)u(t)dt, (5.1) n

0

where u(n) (x) = ddxnu . Because the resulted equation in (5.1) combines the diﬀerential operator and the integral operator, then it is necessary to deﬁne initial conditions u(0), u (0), . . . , u(n−1) (0) for the determination of the particular solution u(x) of the Volterra integro-diﬀerential equation (5.1). Any Volterra integro-diﬀerential equation is characterized by the existence of one or more of the derivatives u (x), u (x),. . . outside the integral sign. The Volterra integro-diﬀerential equations may be observed when we convert an initial value problem to an integral equation by using Leibnitz rule. The Volterra integro-diﬀerential equation appeared after its establishment by Volterra. It then appeared in many physical applications such as glassforming process, nanohydrodynamics, heat transfer, diﬀusion process in general, neutron diﬀusion and biological species coexisting together with increasing and decreasing rates of generating, and wind ripple in the desert. More details about the sources where these equations arise can be found in physics, biology and engineering applications books. To determine a solution for the integro-diﬀerential equation, the initial conditions should be given, and this may be clearly seen as a result of involving u(x) and its derivatives. The initial conditions are needed to determine the exact solution.

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

176

5 Volterra Integro-Diﬀerential Equations

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind In what follows we will present the recently developed methods, namely the Adomian decomposition method (ADM) and the variational iteration method (VIM) that will be used to handle the Volterra integro-diﬀerential equations of the second kind. Moreover, some of the traditional methods, namely the Laplace transform method, the series solution method, converting Volterra integro-diﬀerential equations to equivalent Volterra integral equations and converting Volterra integro-diﬀerential equations to equivalent initial value problem, will be studied as well. However, the Volterra integro-diﬀerential equations of the ﬁrst kind will be examined. The Laplace transform method and the variational iteration method will be used to handle the ﬁrst kind equations.

5.2.1 The Adomian Decomposition Method The Adomian decomposition method [5–9] gives the solution in an inﬁnite series of components that can be recurrently determined. The obtained series may give the exact solution if such a solution exists. Otherwise, the series gives an approximation for the solution that gives high accuracy level. Without loss of generality, we may assume a Volterra integro-diﬀerential equation of the second kind given by x u (x) = f (x) + K(x, t)u(t)dt, u(0) = a0 , u (0) = a1 . (5.2) 0

Integrating both sides of (5.2) from 0 to x twice leads to x K(x, t)u(t)dt , u(x) = a0 + a1 x + L−1 (f (x)) + L−1

(5.3)

0

where the initial conditions u(0) and u (0) are used, and L−1 is a two-fold integral operator. We then use the decomposition series ∞ un (x), (5.4) u(x) = n=0

into both sides of (5.3) to obtain ∞ −1 −1 un (x) = a0 + a1 x + L (f (x)) + L n=0

0

x

K(x, t)

∞

un (t) dt ,

n=0

(5.5) or equivalently

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

u0 (x) + u1 (x) + u2 (x) + u3 (x) + · · · = a0 + a1 x + L−1 (f (x)) x x K(x, t)u0 (t)dt + L−1 K(x, t)u1 (t)dt +L−1 0 0 x +L−1 K(x, t)u2 (t)dt + · · · .

177

(5.6)

0

To determine the components u0 (x), u1 (x), u2 (x), u3 (x), . . . of the solution u(x), we set the recurrence relation u0 (x) = a0 + a1 x + L−1 (f (x)) , x (5.7) uk+1 (x) = L−1 K(x, t)uk (t)dt , k 0, 0

where the zeroth component u0 (x) is deﬁned by all terms not included inside the integral sign of (5.6). Having determined the components ui (x), i 0, the solution u(x) of (5.2) is then obtained in a series form. Using (5.4), the obtained series converges to the exact solution if such a solution exists. n However, for concrete problems, a truncated series k=0 uk (x) is usually used to approximate the solution u(x) that can be used for numerical purposes. Remarks 1. The Adomian decomposition method was presented before to handle a second order Volterra integro-diﬀerential equations. For other orders, we can follow the same approach as presented. This will be explained later in details by discussing the illustrative examples, where ﬁrst-order, second-order, thirdorder, and fourth-order Volterra integro-diﬀerential equations will be studied. 2. The modiﬁed decomposition method that we used before can be used for handling Volterra integro-diﬀerential equations of any order. 3. The phenomenon of the noise terms that was applied before can be used here if noise terms appear. The Adomian decomposition method for solving the second kind Volterra integro-diﬀerential equations will be illustrated by studying the following examples. The selected equations are of orders 1, 2, 3, and 4. Other equations of higher orders can be treated in a like manner. Example 5.1 Use the Adomian method to solve the Volterra integro-diﬀerential equation x u(t)dt, u(0) = 0. (5.8) u (x) = 1 − 0

Applying the integral operator L−1 deﬁned by x −1 L (·) = (·)dx,

(5.9)

0

to both sides of (5.8), i.e. integrating both sides of (5.8) once from 0 to x, and using the given initial condition we obtain

178

5 Volterra Integro-Diﬀerential Equations

u(x) = x − L

−1

x

u(t)dt .

(5.10)

0

Using the decomposition series (5.4), and using the recurrence relation (5.7) we obtain u0 (x) = x, x 1 −1 u1 (x) = −L u0 (t)dt = − x3 , 3! 0 x (5.11) 1 u1 (t)dt = x5 , u2 (x) = L−1 5! 0 x 1 u3 (x) = L−1 u2 (t)dt = − x7 , 7! 0 and so on. This gives the solution in a series form 1 1 1 u(x) = x − x3 + x5 − x7 + · · · , 3! 5! 7! and hence the exact solution is given by u(x) = sin x.

(5.12) (5.13)

Example 5.2 Use the Adomian method to solve the Volterra integro-diﬀerential equation x u (x) = 1 + x + (x − t)u(t)dt, u(0) = 1, u (0) = 1. (5.14) 0

Applying the two-fold integral operator L−1 deﬁned by x x −1 L (·) = (·)dxdx, 0

(5.15)

0

to both sides of (5.14), i.e. integrating both sides of (5.14) twice from 0 to x, and using the given initial conditions we obtain x 1 3 1 2 −1 (x − t)u(t)dt . (5.16) u(x) = 1 + x + x + x + L 2! 3! 0 Using the decomposition series (5.4), and using the recurrence relation (5.7) we obtain 1 1 u0 (x) = 1 + x + x2 + x3 , 2! 3! x (5.17) (x − t)u0 (t)dt u1 (x) = L−1 0

1 1 1 1 4 x + x5 + x6 + x7 , 4! 5! 6! 7! and so on. This gives the solution in a series form 1 1 1 1 1 1 u(x) = 1 + x + x2 + x3 + x4 + x5 + x6 + x7 + · · · , 2! 3! 4! 5! 6! 7! and this converges to the exact solution =

(5.18)

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

u(x) = ex .

179

(5.19)

Example 5.3 Use the Adomian method to solve the Volterra integro-diﬀerential equation x u (x) = −1 + x − (x − t)u(t)dt, u(0) = 1, u (0) = −1, u (0) = 1. 0

Applying the three-fold integral operator L−1 deﬁned by x x x −1 (·)dxdxdx, L (·) = 0

0

(5.20) (5.21)

0

to both sides of (5.20), and using the given initial conditions we obtain x 1 1 1 (x − t)u(t)dt . (5.22) u(x) = 1 − x + x2 − x3 + x4 − L−1 2! 3! 4! 0 Using the decomposition series (5.4), and using the recurrence relation (5.7) we obtain 1 1 1 u0 (x) = 1 − x + x2 − x3 + x4 , 2! 3! 4! x −1 (5.23) (x − t)u0 (t)dt u1 (x) = −L 0

1 1 1 1 1 = − x5 + x6 − x7 + x8 − x9 , 5! 6! 7! 8! 9! and so on. This gives the solution in a series form 1 1 1 1 1 1 u(x) = 1 − x + x2 − x3 + x4 − x5 + x6 − x7 + · · · , 2! 3! 4! 5! 6! 7! and this converges to the exact solution u(x) = e−x .

(5.24) (5.25)

Example 5.4 Use the Adomian method to solve the Volterra integro-diﬀerential equation x (iv) (x − t)u(t)dt, u (x) = −1 + x − (5.26) 0 u(0) = −1, u (0) = 1, u (0) = 1, u (0) = −1. Applying the four-fold integral operator L−1 to both sides of (5.26) and using the given initial condition we obtain x u(x) = x − 1 − L−1 (x − t)u(t)dt . (5.27) 0

Using the decomposition series (5.4), and using the recurrence relation (5.7) we obtain

180

5 Volterra Integro-Diﬀerential Equations

1 2 1 1 1 x − x3 − x4 + x5 , 2! 3! 4! 5! x (5.28) 1 6 1 7 1 8 −1 (x − t)u0 (t)dt = x − x + x + · · · , u1 (x) = L 6! 7! 8! 0 and so on. The series solution is given by 1 1 1 1 1 1 u(x) = x − x3 + x5 − x7 + · · · − 1 − x2 + x4 − x6 + · · · , 3! 5! 7! 2! 4! 6! (5.29) so that the exact solution is given by u0 (x) = −1 + x +

u(x) = sin x − cos x.

(5.30)

Exercises 5.2.1 Solve the following Volterra integro-diﬀerential equations by using the Adomian decomposition method: x u(t)dt, u(0) = 0 1. u (x) = 1 + 0

2. u (x) = 1 + x +

x 0

3. u (x) = 2 + 4x + 8 4. u (x) = 1 +

x

0

5. u (x) = −1 +

(x − t)u(t)dt, u(0) = 1

x

0

(x − t)u(t)dt, u(0) = 1

(x − t)u(t)dt, u(0) = 1, u (0) = 0

x 0

6. u (x) = 1 + x +

(x − t)u(t)dt, u(0) = 1, u (0) = 0

x

0

7. u (x) = −1 − x +

(x − t)u(t)dt, u(0) = 1, u (0) = 1 x

0

(x − t)u(t)dt, u(0) = 1, u (0) = 1

8. u (x) = 2 − 2x sin x − 9. u (x) = −x −

1 3 x + 6

10. u (x) = 1 + x −

x

0 x 0

(x − t)u(t)dt, u(0) = 0, u (0) = 0

(x − t)u(t)dt, u(0) = 0, u (0) = 2

1 3 x + 31

x

0

11. u (x) = 1 − x + 2 sin x −

12. u (x) = 1 + x +

1 3 x + 6

13. u (x) = 1 + x −

1 4 x + 12

(x − t)u(t)dt, u(0) = 1, u (0) = 2 x

0 x

0

(x − t)u(t)dt, u(0) = 1, u (0) = 0, u (0) = 1

x

0

(x − t)u(t)dt, u(0) = 1, u (0) = −1, u (0) = −1

(x − t)u(t)dt, u(0) = 1, u (0) = 1, u (0) = 3

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind 181 x 14. u(iv) (x) = 1 + x − (x − t)u(t)dt, u(0) = u (0) = 1, u (0) = u (0) = −1 0

1 5 x + 15. u(iv) (x) = 1 + x − 20

x

0

(x − t)u(t)dt, u(0) = u (0) = u (0) = 1,

u (0) = 7 16. u(iv) (x) = 1 + x −

1 2 1 x − x3 + 2! 3!

x 0

(x − t)u(t)dt,

u(0) = u (0) = 2, u (0) = u (0) = 1

5.2.2 The Variational Iteration Method In Chapter 3, the variational iteration method was used to handle Volterra integral equations by converting it to an initial value problem or by converting it to an equivalent integro-diﬀerential equation. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the so called Adomian polynomials that we need for nonlinear problems. The standard ith order integro-diﬀerential equation is of the form x (i) u (x) = f (x) + K(x, t)u(t)dt, (5.31) 0 (i)

di u dxi ,

and u(0), u (0), . . . , u(i−1) (0) are the initial conditions. where u (x) = The correction functional for the integro-diﬀerential equation (5.31) is x ξ (i) un+1 (x) = un (x) + λ(ξ) un (ξ) − f (ξ) − K(ξ, r)˜ un (r) dr dξ. 0

0

(5.32) The variational iteration method [8,10] is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ that can be identiﬁed optimally via integration by parts and by using a restricted variation. Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), n 0 of the solution u(x). The zeroth approximation u0 (x) can be any selective function. However, the initial values u(0), u (0), · · · are preferably used for the selective zeroth approximation u0 (x) as will be seen later. Consequently, the solution is given by u(x) = lim un (x). n→∞

(5.33)

It is useful to summarize the Lagrange multipliers as derived in Chapter 3:

182

5 Volterra Integro-Diﬀerential Equations

u + f (u(ξ), u (ξ)) = 0, λ = −1, u + f (u(ξ), u (ξ), u (ξ)) = 0, λ = ξ − x, 1 (ξ − x)2 , 2! u(n) + f (u(ξ), u (ξ), u (ξ), · · · , u(n) (ξ)) = 0,

u + f (u(ξ), u (ξ), u (ξ), u (ξ)) = 0, λ = −

(5.34)

1 (ξ − x)(n−1) . (n − 1)! The VIM will be illustrated by studying the following examples. λ = (−1)n

Example 5.5 Use the variational iteration method to solve the Volterra integro-diﬀerential equation x

u (x) = 1 +

u(t)dt, u(0) = 1.

(5.35)

0

The correction functional for this equation is given by x ξ un+1 (x) = un (x) − un (ξ) − 1 − un (r)dr dξ, 0

(5.36)

0

where we used λ = −1 for ﬁrst-order integro-diﬀerential equations as shown in (5.34). We can use the initial condition to select u0 (x) = u(0) = 1. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1, x ξ u0 (ξ) − 1 − u1 (x) = u0 (x) − u0 (r)dr dξ 0

0

1 = 1 + x + x2 , 2! x u2 (x) = u1 (x) − u1 (ξ) − 1 − 0

0

ξ

u1 (r)dr

dξ

(5.37)

1 2 1 1 x + x3 + x4 , 2! 3! 4! x ξ u2 (r)dr dξ u3 (x) = u2 (x) − u2 (ξ) − 1 − = 1+x+

0

0

1 1 1 1 1 = 1 + x + x2 + x3 + x4 + x5 + x6 , 2! 3! 4! 5! 6! and so on. The VIM admits the use of u(x) = lim un(x),

(5.38)

u(x) = ex .

(5.39)

n→∞

that gives the exact solution

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

183

Example 5.6 Use the variational iteration method to solve the Volterra integro-diﬀerential equation x

u (x) = 1 +

0

(x − t)u(t)dt, u(0) = 1, u (0) = 0.

(5.40)

The correction functional for this equation is given by x ξ (ξ − x) un (ξ) − 1 − (ξ − r)un (r)dr dξ, (5.41) un+1 (x) = un (x) + 0

0

where λ = ξ − x for second-order integro-diﬀerential equations as shown in (5.34). We can use the initial conditions to select u0 (x) = u(0)+xu (0) = 1. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1, x ξ u1 (x) = u0 (x) + (ξ − x) u0 (ξ) − 1 − (ξ − r)u0 (r)dr dξ 0

0

1 1 = 1 + x2 + x4 , 2! 4! x ξ (ξ − x) u1 (ξ) − 1 − (ξ − r)u1 (r)dr dξ u2 (x) = u1 (x) + 0

0

(5.42)

1 2 1 1 1 x + x4 + x6 + x8 , 2! 4! 6! 8! x ξ (ξ − x) u2 (ξ) − 1 − (ξ − r)u2 (r)dr dξ u3 (x) = u2 (x) + = 1+

0

0

1 1 1 1 1 10 1 12 x + x , = 1 + x2 + x4 + x6 + x8 + 2! 4! 6! 8! 10! 12! and so on. The VIM admits the use of u(x) = lim un(x), n→∞

(5.43)

that gives the exact solution u(x) = cosh x.

(5.44)

Example 5.7 Use the variational iteration method to solve the Volterra integro-diﬀerential equation x 1 u (x) = 1 + x + x3 + (x − t)u(t)dt, u(0) = 1, u (0) = 0, u (0) = 1. 3! 0 (5.45) The correction functional for this equation is given by

184

5 Volterra Integro-Diﬀerential Equations

un+1 (x) = un(x) ξ 1 x 1 3 2 − (ξ − x) un (ξ) − 1 − ξ − ξ − (ξ − r)un (r)dr dξ, 2 0 3! 0 (5.46) where λ = − 12 (ξ − x)2 for third-order ODEs and for second-order integrodiﬀerential equations as shown in (5.34). The zeroth approximation u0 (x) can be selected by using the initial con1 2 1 2 ditions, hence we set u0 (x) = u(0) + xu (0) + 2! x u (0) = 1 + 2! x . Using this selection into the correction functional gives the following successive approximations 1 u0 (x) = 1 + x2 , 2! u1 (x) = u0 (x) ξ 1 x 1 ξ3 − − (ξ − x)2 u (ξ − r)u0 (r)dr dξ 0 (ξ) − 1 − ξ − 2 0 3! 0 1 1 1 1 1 1 = 1 + x2 + x3 + x4 + x5 + x6 + x7 , 2! 3! 4! 5! 6! 7! u2 (x) = u1 (x) ξ 1 x 1 3 2 − (ξ − x) u1 (ξ) − 1 − ξ − ξ − (ξ − r)u1 (r)dr dξ 2 0 3! 0 1 1 1 1 1 1 1 = 1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + · · · , 2! 3! 4! 5! 6! 7! 8! (5.47) and so on. The VIM admits the use of u(x) = lim un(x),

(5.48)

u(x) = ex − x.

(5.49)

n→∞

that gives the exact solution

Example 5.8 Use the variational iteration method to solve the Volterra integro-diﬀerential equation x 1 (x − t)u(t)dt, u(iv) (x) = 24 + x + x6 − 30 (5.50) 0 u(0) = 0, u (0) = 1, u (0) = 0, u (0) = −1. The correction functional for this equation is given by un+1 (x) = un (x) ξ 1 1 x ξ6 + (ξ − x)3 u(iv) (ξ − r)un (r)dr dξ, + n (ξ) − 24 − ξ − 6 0 30 0 (5.51)

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

185

where λ = 16 (ξ − x)3 for fourth-order ODEs integro-diﬀerential equations as shown in (5.34). 1 3 Using u0 (x) = x − 3! x into the correction functional gives the following successive approximations 1 u0 (x) = x − x3 , 3! u1 (x) = u0 (x) ξ 1 x 1 6 (iv) 3 + (ξ − x) u0 (ξ) − 24 − ξ − ξ + (ξ − r)u0 (r)dr dξ 6 0 30 0 1 1 1 1 1 x10 , = x4 + x − x3 + x5 − x7 + x9 + 3! 5! 7! 9! 151200 u2 (x) = u1 (x) ξ 1 x 1 (iv) + (ξ − x)3 u1 (ξ) − 24 − ξ − ξ 6 + (ξ − r)u1 (r)dr dξ 6 0 30 0 1 1 1 1 1 11 x + ··· , = x4 + x − x3 + x5 − x7 + x9 − 3! 5! 7! 9! 11! (5.52) and so on. Notice that x10 in u1 (x) has vanished from u2 (x). Similarly other noise terms vanish in the limit. The VIM admits the use of u(x) = lim un(x), n→∞

that gives the exact solution by u(x) = x4 + sin x.

(5.53) (5.54)

Exercises 5.2.2 Solve the following Volterra integro-diﬀerential equations by using the variational iteration method: x 1. u (x) = x − 1 − (x − t)u(t)dt, u(0) = 1 2. u (x) = 1 + x +

0 x 0

(x − t)u(t)dt, u(0) = 1

3. u (x) = 3ex − 2 − x + 4. u (x) = x + 5. u (x) = x −

x

0

x 0

(x − t)u(t)dt, u(0) = 0

(x − t)u(t)dt, u(0) = 0, u (0) = 1

1 3 x + 3!

x

0 x

(x − t)u(t)dt, u(0) = 0, u (0) = 2

1 3 x + (x − t)u(t)dt, u(0) = 1, u (0) = 1 3! 0 x (x − t)u(t)dt, u(0) = 1, u (0) = 1 7. u (x) = 1 + x + 6. u (x) = 1 −

0

186

5 Volterra Integro-Diﬀerential Equations

8. u (x) = −1 − x + 9. u (x) = −1 −

x 0

1 3 x + 3!

(x − t)u(t)dt, u(0) = 1, u (0) = 1

x

0

1 4 x + 12

10. u (x) = 2 − x −

(x − t)u(t)dt, u(0) = 1, u (0) = 1

11. u (x) = 1 − x + 2 sin x −

x

0

(x − t)u(t)dt, u(0) = 0, u (0) = 1 x

0

(x − t)u(t)dt, u(0) = 1, u (0) = −1, u (0) = −1

1 2 1 1 x (x − t)2 u(t)dt, x − x4 + 2! 4! 2 0 u(0) = 1, u (0) = 2, u (0) = 1 1 1 x 13. u (x) = −3 − 2x − x2 + 6ex + (x − t)2 u(t)dt, 2 2 0 u(0) = 0, u (0) = 1, u (0) = 2 x 14. u(iv) (x) = 1 + x − (x − t)u(t)dt, u(0) = u (0) = 1, u (0) = u (0) = −1

12. u (x) = 1 + x +

0

1 1 5 1 x x + 15. u (x) = 1 + x + x2 − (x − t)2 u(t)dt, u(0) = u (0) = u (0) = 1, 2! 60 2 0 u (0) = 3 1 1 1 x 16. u(iv) (x) = 1 + x + x2 − x3 + (x − t)2 u(t)dt, 2! 3 2 0 u(0) = 3, u (0) = u (0) = u (0) = 1 (iv)

5.2.3 The Laplace Transform Method The Laplace transform method was used before for solving Volterra integral equations of the ﬁrst and the second kind in Chapter 3. The details and properties of the Laplace transform method can be found in ordinary diﬀerential equations texts. Before we start applying this method, we summarize some of the concepts presented in Section 1.5. In the Laplace transform convolution theorem, it was stated that if the kernel K(x, t) of the integral equation x u(n) (x) = f (x) + λ K(x, t)u(t)dt, (5.55) 0

depends on the diﬀerence x − t, then it is called a diﬀerence kernel. The integro-diﬀerential equation can thus be expressed as x u(n) (x) = f (x) + λ K(x − t)u(t)dt. (5.56) 0

Consider two functions f1 (x) and f2 (x) that possess the conditions needed for the existence of Laplace transform for each. Let the Laplace transforms for the functions f1 (x) and f2 (x) be given by L{f1 (x)} = F1 (s),

L{f2 (x)} = F2 (s).

(5.57)

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

187

The Laplace convolution product of these two functions is deﬁned by x (f1 ∗ f2 )(x) = f1 (x − t)f2 (t)dt, (5.58)

or (f2 ∗ f1 )(x) =

0

x 0

f2 (x − t)f1 (t)dt.

(5.59)

Recall that (f1 ∗ f2 )(x) = (f2 ∗ f1 )(x).

(5.60)

We can easily show that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x L{(f1 ∗ f2 )(x)} = L f1 (x − t)f2 (t)dt = F1 (s)F2 (s). (5.61) 0

To solve Volterra integro-diﬀerential equations by using the Laplace transform method, it is essential to use the Laplace transforms of the derivatives of u(x). We can easily show that L{u(n) (x)} = sn L{u(x)} − sn−1 u(0) − sn−2 u (0) − · · · − u(n−1) (0). (5.62) This simply gives L{u (x)} = sL{u(x)} − u(0) = sU (s) − u(0), L{u (x)} = s2 L{u(x)} − su(0) − u (0) = s2 U (s) − su(0) − u (0), L{u (x)} = s3 L{u(x)} − s2 u(0) − su (0) − u (0) = s3 U (s) − s2 u(0) − su (0) − u (0),

(5.63)

L{u(iv) (x)} = s4 L{u(x)} − s3 u(0) − s2 u (0) − su (0) − u (0) = s4 U (s) − s3 u(0) − s2 u (0) − su (0) − u (0), and so on for derivatives of higher order. The Laplace transform method can be applied in a similar manner to the approach used before in Chapter 3. We ﬁrst apply the Laplace transform to both sides of (5.56), use the proper Laplace transform for the derivative of u(x), and then solve for U (s). We next use the inverse Laplace transform of both sides of the resulting equation to obtain the solution u(x) of the equation. Recall that Table 1.2 in Chapter 1 should be used. The Laplace transform method for solving Volterra integro-diﬀerential equations will be illustrated by studying the following examples. Example 5.9 Use the Laplace transform method to solve the Volterra integro-diﬀerential equation x u(t)dt, u(0) = 1. (5.64) u (x) = 1 + 0

188

5 Volterra Integro-Diﬀerential Equations

Notice that the kernel K(x − t) = 1. Taking Laplace transform of both sides of (5.64) gives L(u (x)) = L(1) + L(1 ∗ u(x)), (5.65) so that

1 1 + U (s), (5.66) s s obtained upon using (5.63). Using the given initial condition and solving for U (s) we ﬁnd 1 . (5.67) U (s) = s−1 sU (s) − u(0) =

By taking the inverse Laplace transform of both sides of (5.67), the exact solution is given by u(x) = ex . (5.68) Example 5.10 Use the Laplace transform method to solve the Volterra integro-diﬀerential equation x (x − t)u(t)dt, u(0) = 1, u (0) = 1. (5.69) u (x) = −1 − x + 0

Notice that the kernel K(x − t) = (x − t). Taking Laplace transform of both sides of (5.69) gives (5.70) L(u (x)) = −L(1) − L(x) + L((x − t) ∗ u(x)), so that

1 1 1 (5.71) s2 U (s) − su(0) − u (0) = − − 2 + 2 U (s), s s s obtained upon using (5.63). Using the given initial condition and solving for U (s) we ﬁnd 1 s U (s) = 2 + . (5.72) s + 1 s2 + 1 By taking the inverse Laplace transform of both sides of (5.72), the exact solution is given by u(x) = sin x + cos x. (5.73)

Example 5.11 Use the Laplace transform method to solve the Volterra integro-diﬀerential equation 1 2 1 4 1 x (x− t)2 u(t)dt, u(0) = 1, u (0) = 2, u = 1. u (x) = 1 + x+ x − x + 2! 4! 2 0 (5.74) Taking Laplace transform of both sides of (5.74) gives 1 1 1 L{u (x)} = L{1}+L{x}+ L{x2 }− L{x4 }+ L{(x−t)2 ∗u(x)}, (5.75) 2! 4! 2

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

189

so that

1 1 1 1 1 + + 3 − 5 + 3 U (s). s s2 s s s Using the given initial condition and solving for U (s) we ﬁnd 1 1 U (s) = 2 + . s s−1 Using the inverse Laplace transform, the exact solution is given by u(x) = x + ex . s3 U (s) − s2 − 2s − 1 =

(5.76)

(5.77) (5.78)

Example 5.12 Solve the Volterra integral equation by using by the Laplace transform method x sin(x − t)u(t)dt, u(iv) (x) = sin x + cos x + 2 (5.79) 0 u(0) = u (0) = u (0) = u (0) = 1. Taking Laplace transform of both sides of (5.79) gives L{u(iv) (x)} = L{sin x} + L{cos x} + 2L{sin(x − t) ∗ u(x)},

(5.80)

so that 1 s 2 + + U (s). s2 + 1 s2 + 1 s2 + 1 Using the given initial condition and solving for U (s) we ﬁnd 1 . U (s) = s−1 Using the inverse Laplace transform, the exact solution is given by u(x) = ex . s4 U (s) − s3 − s2 − s − 1 =

(5.81)

(5.82) (5.83)

Exercises 5.2.3 Solve the following Volterra integro-diﬀerential equations by using the Laplace transform method x 1. u (x) = 1 + x − x2 + (x − t)u(t)dt, u(0) = 3 0

1 2. u (x) = 2 + x − x3 + 3!

x 0

3. u (x) = −(a + b)u(x) − ab 4. u (x) = −x +

x

(x − t)u(t)dt, u(0) = 1

x 0

u(t)dt, u(0) = a − b

(x − t)u(t)dt, u(0) = 0, u (0) = 1 x 1 5. u (x) = −x − x3 + (x − t)u(t)dt, u(0) = 0, u (0) = 2 3! 0 x e−(x−t)u(t)dt, u(0) = 1, u (0) = 1 6. u (x) = cosh x + 0

0

190

5 Volterra Integro-Diﬀerential Equations

7. u (x) = cosh x − 8. u (x) = 4ex + 4

x 0 x

0

e(x−t) u(t)dt, u(0) = 1, u (0) = −1

e(x−t) u(t)dt, u(0) = 1, u (0) = 2

1 3 1 x x − (x − t)3 u(t)dt, u(0) = 0, u (0) = 1 3! 6 0 1 1 x (x − t)3 u(t)dt, u(0) = 1, u (0) = 0 10. u (x) = 1 + x2 + 2 6 0 x 11. u (x) = 1 + x − 2x2 + (x − t)u(t)dt, u(0) = 5, u (0) = u (0) = 1 9. u (x) = −x +

1 1 12. u (x) = 1 + x2 + 2! 2

0 x

0

(x − t)2 u(t)dt, u(0) = u (0) = 0, u (0) = 1

1 4 1 x x + (x − t)2 u(t)dt, u(0) = u (0) = u (0) = 1 4! 2 0 x 14. u(iv) (x) = e2x − e2(x−t) u(t)dt, u(0) = u (0) = u (0) = u (0) = 1 13. u (x) = x −

0

15. u(iv) (x) = 3ex + e2x −

x

0

e2(x−t) u(t)dt, u(0) = 0,

u (0) = 1, u (0) = 2, u (0) = 3 x 1 5 1 16. u(iv) (x) = − − x + e2x − e2(x−t) u(t)dt, 4 2 4 0 u(0) = 1, u (0) = 2, u (0) = u (0) = 1

5.2.4 The Series Solution Method It was stated before in Chapter 3 that a real function u(x) is called analytic if it has derivatives of all orders such that the Taylor series at any point b in its domain ∞ u(n) (b) (x − b)n , (5.84) u(x) = n! n=0 converges to u(x) in a neighborhood of b. For simplicity, the generic form of Taylor series at x = 0 can be written as ∞ a n xn . (5.85) u(x) = n=0

In this section we will apply the series solution method for solving Volterra integro-diﬀerential equations of the second kind. We will assume that the solution u(x) of the Volterra integro-diﬀerential equation x u(n) (x) = f (x)+λ K(x, t)u(t)dt, u(k) (0) = k!ak , 0 k (n−1), (5.86) 0

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

191

is analytic, and therefore possesses a Taylor series of the form given in (5.85), where the coeﬃcients an will be determined recurrently. The ﬁrst few coeﬃcients ak can be determined by using the initial conditions so that 1 1 a0 = u(0), a1 = u (0), a2 = u (0), a3 = u (0), (5.87) 2! 3! and so on. The remaining coeﬃcients ak of (5.85) will be determined by applying the series solution method to the Volterra integro-diﬀerential equation (5.86). Substituting (5.85) into both sides of (5.86) gives ∞ (n) ∞ x k k (5.88) ak x = T (f (x)) + K(x, t) ak t dt, 0

k=0

or for simplicity we use (a0 +a1 x+a2 x2 +· · · )(n) = T (f (x))+λ

k=0

x 0

K(x, t) a0 + a1 t + a2 t2 + · · · dt,

(5.89) where T (f (x)) is the Taylor series for f (x). The integro-diﬀerential equation (5.86) will be converted to a traditional integral in (5.88) or (5.89) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integral in (5.88) or (5.89), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0, where some of these coeﬃcients will be used from the initial conditions. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (5.85). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. Example 5.13 Use the series solution method to solve the Volterra integro-diﬀerential equation x u (x) = 1 +

u(t)dt.

(5.90)

0

Substituting u(x) by the series u(x) =

∞ n=0

into both sides of Eq. (5.90) leads to

an xn ,

(5.91)

192

5 Volterra Integro-Diﬀerential Equations

∞

n

an x

x

=1+ 0

n=0

∞

an t

n

dt.

(5.92)

n=0

Diﬀerentiating the left side once with respect to x, and evaluating the integral at the right side we ﬁnd ∞ ∞ 1 n−1 an−1 xn , nan x =1+ (5.93) n n=1 n=1 that can be rewritten as ∞ ∞ 1 n an−1 xn , (n + 1)an+1 x = 1 + a1 + n n=1 n=1

(5.94)

where we uniﬁed the exponent of x in both sides, and used a0 = 0 from the given initial condition. Equating the coeﬃcients of like powers of x in both sides of (5.94) gives the recurrence relation 1 an−1 , n 1. a0 = 0, a1 = 1, an+1 = (5.95) n(n + 1) where this result gives 1 a2n = 0, a2n+1 = , (5.96) (2n + 1)! for n 0. Substituting this result into (5.91) gives the series solution ∞ 1 (5.97) x2n+1 , u(x) = (2n + 1)! n=0 that converges to the exact solution u(x) = sinh x.

(5.98)

Example 5.14 Use the series solution method to solve the Volterra integro-diﬀerential equation x u (x) = 1 + x +

0

(x − t)u(t)dt,

u(0) = u (0) = 1.

(5.99)

Substituting u(x) by the series u(x) =

∞

an xn ,

(5.100)

n=0

into both sides of Eq. (5.99) leads to ∞ x ∞ n n (x − t) dt. an x =1+x+ an t n=0

0

(5.101)

n=0

Diﬀerentiating the left side twice, and by evaluating the integral at the right side we ﬁnd ∞ ∞ 1 an xn+2 , n(n − 1)an xn−2 = 1 + x + (5.102) (n + 2)(n + 1) n=2 n=0

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

or equivalently ∞ ∞ 2a2 +6a3 x+ (n+2)(n+1)an+2 xn = 1+x+ n=2

n=2

193

1 an−2 xn . (5.103) n(n − 1)

Using the initial conditions and equating the coeﬃcients of like powers of x in both sides of (5.103) gives the recurrence relation 1 1 a0 = 1, a1 = 1, a2 = , a3 = , 2! 3! (5.104) 1 an+2 = an−2 , n 2. (n + 2)(n + 1)n(n − 1) where this result gives 1 an = , (5.105) n! for n 0. Substituting this result into (5.100) gives the series solution ∞ 1 n x , (5.106) u(x) = n! n=0 that converges to the exact solution u(x) = ex .

(5.107)

Example 5.15 Use the series solution method to solve the Volterra integro-diﬀerential equation u (x) = 1 − x + 2 sin x −

x

0

(x − t)u(t)dt, u(0) = 1, u (0) = u (0) = −1.

(5.108)

Substituting u(x) by the series u(x) =

∞

an xn ,

(5.109)

n=0

into both sides of the equation (5.108), and using Taylor expansion for sin x we obtain ∞ x ∞ ∞ (−1)n 2n+1 n n an x =1−x+2 − an t (x − t) dt. x (2n + 1)! 0 n=0 n=0 n=0 (5.110) Diﬀerentiating the left side three times, and by evaluating the integral at the right side we ﬁnd 1 1 1 2 3 2 6a3 + 24a4x + 60a5 x + 120a6 x + · · · = 1 + x − a0 x − + a1 x3 + · · · , 2 3 6 (5.111) where we used few terms for simplicity reasons. Using the initial conditions and equating the coeﬃcients of like powers of x in both sides of (5.111) we ﬁnd

194

5 Volterra Integro-Diﬀerential Equations

a0 = 1,

a1 = −1,

1 , 2! 1 a5 = − . 5! a2 = −

1 1 , a4 = , 3! 4! Consequently, the series solution is given by ∞ ∞ (−1)n 2n (−1)n 2n+1 u(x) = x − x , (2n)! (2n + 1)! n=0 n=0 a3 =

that converges to the exact solution u(x) = cos x − sin x.

(5.112)

(5.113)

(5.114)

Example 5.16 Use the series solution method to solve the Volterra integro-diﬀerential equation u(iv) (x) = 1 + x −

x

0

(x − t)u(t)dt, u(0) = u (0) = 1, u (0) = u (0) = −1.

(5.115)

Proceeding as before we set (iv) ∞ n an x =1+x−

0

n=0

x

(x − t)

∞

an t

n

dt,

(5.116)

n=0

Diﬀerentiating the left side, integrating the right side, and using few terms for simplicity we ﬁnd 1 1 24a4 +120a5x+360a6x2 +840a7x3 +· · · = 1+x− a0 x2 − a1 x3 +· · · . (5.117) 2 6 Using the initial conditions and equating the coeﬃcients of like powers of x in both sides of (5.117) gives the recurrence relation 1 a0 = 1, a1 = 1, a2 = − , 2! (5.118) 1 1 1 1 a6 = − , · · · a3 = − , a 4 = , a 5 = , 3! 4! 5! 6! Consequently, the series solution is given by ∞ ∞ (−1)n 2n (−1)n 2n+1 , (5.119) u(x) = x + x (2n)! (2n + 1)! n=0 n=0 that converges to the exact solution u(x) = cos x + sin x.

(5.120)

Exercises 5.2.4 Solve the following Volterra integro-diﬀerential equations by using the series solution method: x 1. u (x) = 1 + u(t)dt, u(0) = 0 0

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind x 2. u (x) = 1 + x + (x − t)u(t)dt, u(0) = 1

195

0

x 1 3. u (x) = 2 − x + x3 − (x − t)u(t)dt, u(0) = −1 3! 0 x 4. u (x) = 1 + (x − t)u(t)dt, u(0) = 1, u (0) = 0 0

x 1 (x − t)u(t)dt, u(0) = 0, u (0) = 2 5. u (x) = x − x3 + 3! 0 x 6. u (x) = cosh x + e−(x−t)u(t)dt, u(0) = 1, u (0) = 1

0

7. u (x) = −1 − x +

x 0

(x − t)u(t)dt, u(0) = 1, u (0) = 1

x 1 4 x + (x − t)u(t)dt, u(0) = 0, u (0) = 1 12 0 1 1 x 9. u (x) = −x + x3 − (x − t)3 u(t)dt, u(0) = 0, u (0) = 1 3! 6 0 x 1 (x − t)u(t)dt, u(0) = 1, u (0) = 2 10. u (x) = 1 + x − x3 + 3! 0 x 1 11. u (x) = 1 + x + x3 + (x − t)u(t)dt, u(0) = 1, u (0) = 0, u (0) = 1 3! 0 1 1 x 12. u (x) = 1 + x2 + (x − t)2 u(t)dt, u(0) = u (0) = 0, u (0) = 1 2! 2 0 x 1 4 x + 13. u (x) = 1 + x − (x − t)u(t)dt, u(0) = 1, u (0) = 1, u (0) = 3 12 0 x 1 5 x + 14. u(iv) (x) = 1 + x − (x − t)u(t)dt, 20 0 u(0) = u (0) = u (0) = 1, u (0) = 7 x 15. u(iv) (x) = 3ex + e2x − e2(x−t) u(t)dt, u(0) = 0,

8. u (x) = 2 − x −

0

u (0) = 1, u (0) = 2, u (0) = 3

x 1 2 1 (x − t)u(t)dt, x − x3 + 2! 3! 0 u(0) = u (0) = 2, u (0) = u (0) = 1

16. u(iv) (x) = 1 + x −

5.2.5 Converting Volterra Integro-Diﬀerential Equations to Initial Value Problems The Volterra integro-diﬀerential equation x u(n) (x) = f (x)+λ K(x, t)u(t)dt, u(k) (0) = k!ak , 0 k (n−1), (5.121) 0

196

5 Volterra Integro-Diﬀerential Equations

can be solved by converting it to an equivalent initial value problem. The study will be concerned on the Volterra integro-diﬀerential equations where the kernel is a diﬀerence kernel of the form K(x, t) = K(x − t), such as x − t. The reason for this selection is that we seek ODEs with constant coeﬃcients. Having converted the integro-diﬀerential equation to an initial value problem, we then can use any standard method for solving ODEs. It is worth noting that the conversion process can be easily used, but it requires more works if compared with the integro-diﬀerential equations methods. The conversion process is obtained by diﬀerentiating both sides of the Volterra integro-diﬀerential equation as many times until we get rid of the integral sign. To perform the diﬀerentiation for the integral at the right side, the Leibnitz rule, that was introduced in Chapter 1, should be used. The initial conditions should be determined by using a variety of integral equations that we will obtain in the process of diﬀerentiation. To give a clear overview of this method we discuss the following illustrative examples. Example 5.17 Solve the Volterra integro-diﬀerential equation by converting it to an initial value problem x u (x) = 1 + u(t)dt, u(0) = 1. (5.122) 0

Diﬀerentiating both sides of (5.122) with respect to x and using the Leibnitz rule we ﬁnd u (x) = u(x), (5.123) with initial conditions given by (5.124) u(0) = 1, u (0) = 1. The second initial condition is obtained by substituting x = 0 in both sides of (5.122). The characteristic equation for the ODE (5.122) is r2 − 1 = 0, (5.125) which gives the roots r = ±1 so that the general solution is given by u(x) = Aex + Be−x .

(5.126) (5.127)

The constants A and B can be determined by using the initial conditions, where we ﬁnd A = 1, B = 0. The exact solution is (5.128) u(x) = ex . Example 5.18 Solve the Volterra integro-diﬀerential equation by converting it to an initial value problem x u (x) = 6 − 3x2 + u(t)dt, u(0) = 0. (5.129) 0

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

197

Diﬀerentiating both sides of (5.129) with respect to x and using the Leibnitz rule we ﬁnd u (x) − u(x) = −6x, (5.130) with initial conditions given by u(0) = 0, u (0) = 6. (5.131) The second initial condition is obtained by substituting x = 0 in both sides of (5.129). The characteristic equation for the related homogeneous ODE of (5.129) is r2 − 1 = 0, (5.132) which gives the roots r = ±1

(5.133)

so that the complementary solution is given by uc (x) = Aex + Be−x . To ﬁnd a particular solution up (x), we substitute up (x) = C + Dx,

(5.134) (5.135)

into (5.130) and equate coeﬃcients of like powers of x from both side to ﬁnd that C = 0, D = 6. This gives the general solution u(x) = uc (x) + up (x) = Aex + Be−x + 6x. (5.136) The constants A and B can be determined by using the initial conditions, where we ﬁnd A = 0, B = 0. The exact solution is given by u(x) = 6x. (5.137) Example 5.19 Solve the following Volterra integro-diﬀerential equation by converting it to an initial value problem x u (x) = 1 + x + (x − t)u(t)dt, u(0) = 1, u (0) = 1. (5.138) 0

Diﬀerentiating both sides and proceeding as before we ﬁnd x u(t)dt, u (x) = 1 +

(5.139)

0

and by diﬀerentiating again to remove the integral sign we ﬁnd u(iv) (x) = u(x).

(5.140) Two more initial conditions can be obtained by substituting x = 0 in (5.138) and (5.139), hence the four initial conditions are (5.141) u(0) = 1, u (0) = 1, u (0) = 1, u (0) = 1. The characteristic equation of (5.140) is r4 − 1 = 0, which gives the roots r = ±1, ±i, i =

√ −1,

(5.142) (5.143)

198

5 Volterra Integro-Diﬀerential Equations

so that the general solution is given by (5.144) u(x) = Aex + Be−x + C cos x + D sin x. The constants A, B, C, and D can be obtained by using the initial conditions where we found A = 1 and B = C = D = 0. The exact solution is given by u(x) = ex .

(5.145)

Example 5.20 Solve the following Volterra integro-diﬀerential equation by converting it to an initial value problem x (x − t)u(t)dt, u(0) = 1, u (0) = 1. (5.146) u (x) = −1 − x + 0

Diﬀerentiating both sides and proceeding as before we obtain x u (x) = −1 + u(t)dt,

(5.147)

0

and by diﬀerentiating again to remove the integral sign we ﬁnd u(iv) (x) = u(x).

(5.148)

Two more initial conditions can be obtained by substituting x = 0 in (5.146) and (5.147), hence the four initial conditions are u(0) = 1, u (0) = 1, u (0) = −1, The characteristic equation of (5.148) is

u (0) = −1.

r4 − 1 = 0, which gives the roots r = ±1, ±i, i =

√ −1,

(5.149) (5.150) (5.151)

so that the general solution is given by u(x) = Aex + Be−x + C cos x + D sin x. (5.152) The constants A, B, C, and D can be obtained by using the initial conditions where we found A = B = 0 and C = D = 1. The exact solution is given by u(x) = sin x + cos x.

(5.153)

Exercises 5.2.5 Solve the following Volterra integro-diﬀerential equations by converting the problem to an initial value problem: x 1. u (x) = 1 + u(t)dt, u(0) = 0 0

2. u (x) = −1 + x − 3. u (x) = 1 +

x 0

x

0

(x − t)u(t)dt, u(0) = 1

(x − t)u(t)dt, u(0) = 1, u (0) = 0

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind x 4. u (x) = x + (x − t)u(t)dt, u(0) = 0, u (0) = 1

199

0

x 1 3 x + (x − t)u(t)dt, u(0) = 1, u (0) = 1 3! 0 x 6. u (x) = −x + (x − t)u(t)dt, u(0) = 0, u (0) = 1 5. u (x) = −1 −

0

7. u (x) = 1 + x −

1 3 x + 3!

1 2 8. u (x) = − x2 − x3 + 2 3

x 0

x

0

(x − t)u(t)dt, u(0) = 1, u (0) = 2 (x − t)u(t)dt, u(0) = 1, u (0) = 4

5.2.6 Converting Volterra Integro-Diﬀerential Equation to Volterra Integral Equation The Volterra integro-diﬀerential equation x (n) u (x) = f (x) + λ K(x, t)u(t)dt,

(5.154)

0

can also be solved by converting it to an equivalent Volterra integral equation. Recall that in integro-diﬀerential equations, the initial conditions are usually prescribed. The study will be focused on the Volterra integro-diﬀerential equations where the kernel is a diﬀerence kernel. Having converted the integro-diﬀerential equation to an equivalent integral equation, the latter can be solved by using any of the methods presented in Chapter 3, such as Adomian method, series solution method, and Laplace transform method. It is obvious that the Volterra integro-diﬀerential equation (5.154) involves derivatives at the left side, and integral at the right side. To perform the conversion process, we need to integrate both sides n times to convert it to a standard Volterra integral equation. It is therefore useful to summarize some of formulas to support the conversion process. We point out that the ﬁrst set of formulas is usually studied in calculus. However, the second set is given in Section 1.4, where reducing multiple integrals to a single integral is presented. I. Integration of derivatives: from calculus we observe the following: x u (t)dt = u(x) − u(0), 0

x

0

x

0

x1

0 0 x1 x 2 0

u (t)dtdx1 = u(x) − xu (0) − u(0),

u (t)dtdx2 dx1 = u(x) −

and so on for other derivatives.

1 2 x u (0) − xu (0) − u(0), 2!

(5.155)

200

5 Volterra Integro-Diﬀerential Equations

II. Reducing multiple integrals to a single integral: from Chapter 1, we studied the following: x x1 x u(t)dtdx1 = (x − t)u(t)dt, 0 x

0 x1

0

0 x 0

0 x x1

0

x1

0

0

1 (x − t)u(t)dtdx1 = 2

1 (x − t) u(t)dtdx1 = 3 2

(x − t)3 u(t)dtdx1 =

1 4

x

0

0

(5.156)

x

3

(x − t) u(t)dt,

0

(x − t)2 u(t)dt,

x

(x − t)4 u(t)dt,

and so on. This can be generalized in the form xn−1 x x1 x 2 1 x ··· (x − t)u(t)dtdxn−1 · · · dx1 = (x − t)n u(t)dt. n! 0 0 0 0 0 (5.157) The conversion to an equivalent Volterra integral equation will be illustrated by studying the following examples. Example 5.21 Solve the following Volterra integro-diﬀerential equation by converting it to Volterra integral equation x u(t)dt, u(0) = 0. (5.158) u (x) = 1 + 0

Integrating both sides from 0 to x, and using the aforementioned formulas we ﬁnd x x u(x) − u(0) = x + u(t)dt. (5.159) 0

0

Using the initial condition gives the Volterra integral equation x (x − t)u(t)dt. u(x) = x +

(5.160)

0

We can select any of the proposed methods. The Adomian decomposition method will be used to solve this problem. Using the series assumption ∞ u(x) = un (x), (5.161) n=0

into both sides of the Volterra integral equation gives the recursion relation 1 1 1 (5.162) u0 (x) = x, u1 (x) = x3 , u2 (x) = x5 , u3 (x) = x7 , 3! 5! 7! so that the series solution is given by 1 1 1 (5.163) u(x) = x + x3 + x5 + x7 + · · · , 3! 5! 7! that converges to the exact solution

5.2 Volterra Integro-Diﬀerential Equations of the Second Kind

u(x) = sinh x.

201

(5.164)

Example 5.22 Solve the following Volterra integro-diﬀerential equation by converting it to Volterra integral equation x (x − t)u(t)dt, u(0) = 3. (5.165) u (x) = 1 + x − x2 + 0

Integrating both sides once from 0 to x and using the formulas given above we obtain 1 2 1 3 1 x u(x) − u(0) = x + x − x + (x − t)2 u(t)dt. (5.166) 2 3 2! 0 Notice that when we integrate the integral at the right side, we should use the formula (5.157) for the resulting double integral. By using the initial condition we obtain the Volterra integral equation 1 x 1 2 1 3 (x − t)2 u(t)dt. (5.167) u(x) = 3 + x + x − x + 2 3 2! 0 We proceed as before and use the Adomian decomposition method to ﬁnd 1 1 u0 (x) = 3 + x + x2 − x3 , 2 3 1 1 1 1 6 (5.168) u1 (x) = x3 + x4 + x5 − x , 2! 4! 5! 360 1 6 1 1 x + x7 + x8 + · · · . u2 (x) = 240 7! 8! Consequently, the series solution is given by 1 3 1 4 1 5 1 2 (5.169) u(x) = 2 + 1 + x + x + x + x + x + · · · , 2! 3! 4! 5! that converges to the exact solution u(x) = 2 + ex . (5.170) Example 5.23 Solve the following Volterra integro-diﬀerential equation by converting it to Volterra integral equation x (x − t)u(t)dt, u(0) = 0, u (0) = 1. (5.171) u (x) = −x + 0

Integrating both sides twice from 0 to x and using the formulas given above we ﬁnd 1 3 1 x (x − t)3 u(t)dt. (5.172) u(x) − xu (0) − u(0) = − x + 3! 3! 0 Notice that when we integrate the integral at the right side, we should use the formula (5.157) for the resulting triple integral. By using the initial conditions we obtain the Volterra integral equation

202

5 Volterra Integro-Diﬀerential Equations

1 3 1 x u(x) = x − x + (x − t)3 u(t)dt. (5.173) 3! 3! 0 We now select the successive approximations method and set the recurrence relation 1 1 x un (x) = x − x3 + (x − t)3 un−1 (t)dt, n 1. (5.174) 3! 3! 0 Selecting the zeroth approximation u0 (x) = x gives the approximations u0 (x) = x, 1 3 1 x + x5 , 3! 5! 1 1 1 1 u2 (x) = x − x3 + x5 − x7 + x9 . 3! 5! 7! 9! Consequently, the series solution is given by 1 1 1 1 u(x) = x − x3 + x5 − x7 + x9 + · · · , 3! 5! 7! 9! that converges to the exact solution u(x) = sin x. u1 (x) = x −

(5.175)

(5.176) (5.177)

Example 5.24 Solve the following Volterra integro-diﬀerential equation by converting it to Volterra integral equation x u (x) = 1 − x + 2 sin x − (x − t)u(t)dt, u(0) = 1, u (0) = −1, u (0) = −1. 0

(5.178) Integrating both sides three times from 0 to x and using the formulas given above we ﬁnd 1 3 1 4 1 x 1 2 2 (x − t)4 u(t)dt. u(x) + x + x − 1 = x − x + x + 2 cos x − 2 + 2! 3! 4! 4! 0 (5.179) By using the initial conditions we obtain the Volterra integral equation 1 1 1 x 1 (x − t)4 u(t)dt. (5.180) u(x) = −1 − x + x2 + x3 − x4 + 2 cos x + 2! 3! 4! 4! 0 Using the Adomian decomposition method gives the recurrence relation 1 1 1 u0 (x) = −1 − x + x2 + x3 − x4 + 2 cos x, 2! 3! 4! 1 3 1 5 1 6 (5.181) u1 (x) = 2x − x + x + x − 2 sin x + · · · , 3 5! 6! 1 1 6 x − 2 cos x + · · · , u2 (x) = 2 − x2 + x4 − 12 360 and so on. Consequently, the series solution is given by

5.3 Volterra Integro-Diﬀerential Equations of the First Kind

203

1 2 1 4 1 3 1 5 u(x) = 1 − x + x + · · · + x − x + x − · · · − 2 sin x, 2! 4! 3! 5! (5.182) that converges to the exact solution u(x) = cos x − sin x. (5.183)

Exercises 5.2.6 Solve the following Volterra integro-diﬀerential equations by converting the problem to Volterra integral equation: x x u(t)dt, u(0) = 0 2. u (x) = 1 − u(t)dt, u(0) = 1 1. u (x) = 1 − 0

3. u (x) = x +

0

x 0

4. u (x) = 1 + x + 5. u (x) = −1 −

(x − t)u(t)dt, u(0) = 0, u (0) = 1

x 0

(x − t)u(t)dt, u(0) = u (0) = 1

1 3 x + 3!

6. u (x) = 1 + x − 7. u (x) = 1 + x +

x

0

1 3 x + 3!

x 0

1 3 x + 3!

8. u (x) = 1 − e−x +

(x − t)u(t)dt, u(0) = 0, u (0) = 2

(x − t)u(t)dt, u(0) = 1, u (0) = 2

x

0

1 3 x + 3!

(x − t)u(t)dt, u(0) = u (0) = 1, u (0) = 0

x 0

(x − t)u(t)dt, u(0) = u (0) = 1, u (0) = 0

5.3 Volterra Integro-Diﬀerential Equations of the First Kind The standard form of the Volterra integro-diﬀerential equation [11–12] of the ﬁrst kind is given by x x K1 (x, t)u(t)dt + K2 (x, t)u(n) (t)dt = f (x), K2 (x, t) = 0, (5.184) 0

0

where initial conditions are prescribed. The Volterra integro-diﬀerential equation of the ﬁrst kind (5.184) can be converted into a Volterra integral equation of the second kind, for n = 1, by integrating the second integral in (5.184) by parts. The Volterra integro-diﬀerential equations of the ﬁrst kind will be handled in this section by Laplace transform method and the variational iteration method.

204

5 Volterra Integro-Diﬀerential Equations

5.3.1 Laplace Transform Method The Laplace transform method was used before for solving Volterra integral equations of the ﬁrst and the second kind in Chapter 3. It was also used in this chapter for solving integro-diﬀerential equations of the second kind. The analysis will be focused on equations where the kernels K1 (x, t) and K2 (x, t) of (5.184) are diﬀerence kernels. This means that each kernel depends on the diﬀerence (x − t). Taking Laplace transform of both sides of (5.184) gives (5.185) L(K1 (x − t) ∗ u(x)) + L(K2 (x − t) ∗ u(n) (x)) = L(f (x)), so that K1 (s)U (s)+K2 (s) sn U (s) − sn−1 u(0) − sn−2 u (0) − · · · − u(n−1) (0) = F (s), (5.186) where U (s) = L(u(x)), K1 (s) = L(K1 (x)),

K2 (s) = L(K2 (x)),

F (s) = L(f (x)). (5.187) Using the given initial conditions and solving for U (s) we ﬁnd

F (s) + K2 (s) sn−1 u(0) + sn−2 u (0) + · · · − u(n−1) (0) , (5.188) U (s) = K1 (s) + sn K2 (s) provided that (5.189) K1 (s) + sn K2 (s) = 0. By taking the inverse Laplace transform of both sides of (5.188), the exact solution is readily obtained. The analysis presented above can be explained by using the following illustrative examples. Example 5.25 Solve the following Volterra integro-diﬀerential equation of the ﬁrst kind x x (x − t)u(t)dt + (x − t)2 u (t)dt = 3x − 3 sin x, u(0) = 0. (5.190) 0

0

Taking Laplace transforms of both sides gives 1 2 3 3 U (s) + 3 (sU (s) − u(0)) = 2 − , (5.191) 2 s s s 1 + s2 where by using the given initial condition and solving for U (s) we obtain 1 U (s) = . (5.192) 1 + s2 Taking the inverse Laplace transform of both sides we ﬁnd u(x) = sin x. (5.193) Example 5.26 Solve the following Volterra integro-diﬀerential equation of the ﬁrst kind

5.3 Volterra Integro-Diﬀerential Equations of the First Kind

0

x

(x − t + 1)u (t)dt = 2xex + ex − x − 1,

u(0) = 0,

205

u (0) = 1. (5.194)

Notice in this equation that K1 (x, t) = 0, and K2 (x, t) = (x − t + 1). Taking Laplace transforms of both sides gives 1 2 1 1 1 1 + + (s2 U (s) − su(0) − u (0)) = − 2 − , (5.195) 2 2 s s (s − 1) s−1 s s where by using the given initial conditions and solving for U (s) we obtain 1 . (5.196) U (s) = (s − 1)2 Taking the inverse Laplace transform of both sides we ﬁnd u(x) = xex . (5.197) Example 5.27 Solve the following Volterra integro-diﬀerential equation of the ﬁrst kind x x cos(x − t)u(t)dt + sin(x − t)u (t)dt = 1 + sin x − cos x, (5.198) 0 0 u(0) = 1, u (0) = 1, u (0) = −1. Notice in this equation that K1 (x, t) = cos(x − t), and K2 (x, t) = sin(x − t). Taking Laplace transforms of both sides gives 1 1 s 1 s U (s) + (s3 U (s) − s2 u(0) − su(0) − u (0)) = + − , 2 2 2 1+s 1+s s 1+s 1 + s2 (5.199) where by using the given initial conditions and solving for U (s) we obtain 1 s U (s) = 2 + . (5.200) s 1 + s2 Taking the inverse Laplace transform of both sides we ﬁnd u(x) = x + cos x.

(5.201)

Example 5.28 Solve the following Volterra integro-diﬀerential equation of the ﬁrst kind x 1 x 1 (x − t)u(t)dt + (x − t − 1)u (t)dt = sin 2x, u(0) = 1, u (0) = 0. 4 0 2 0 (5.202) Taking Laplace transforms of both sides gives

1 1 1 2 1 1 s U (s) − su(0) − u (0) = 2 , (5.203) U (s) + − 2 2 s 4 s s s +4 where by using the given initial conditions and solving for U (s) we obtain s U (s) = . (5.204) 4 + s2 Taking the inverse Laplace transform of both sides we ﬁnd u(x) = cos 2x. (5.205)

206

5 Volterra Integro-Diﬀerential Equations

Exercises 5.3.1 Solve the following Volterra integro-diﬀerential equations of the ﬁrst kind by using Laplace transform method: x x 1 1. (x − t)u(t)dt + (x − t)2 u (t)dt = 3x + x2 − 3 sin x, u(0) = 0 2 0 0 x x (x − t)u(t)dt + (x − t)2 u (t)dt = 6 − 6 cos x − 3x sin x, u(0) = 0 2.

0 x

3. 0 x

4. 0 x

5. 0 x

6.

0 x

7. 0 x

8. 0

0

1 (x − t)u(t)dt − 2

x 0

(x − t + 1)u (t)dt = 2x2 −

(x − t + 1)u (t)dt = ex +

(x − t)u(t)dt +

x

0

sin(x − t)u(t)dt −

(x − t + 1)u (t)dt = 1 2

cosh(x − t)u(t)dt +

0

x

10. 0 x

11. 0 x

12. 0

1 2 x − 1, u(0) = 1 2

(x − t + 1)u (t)dt = 2 sin x − x, u(0) = −1, u (0) = 1

x

0 x

0

u(0) = 0, u (0) = 1 x cosh(x − t)u(t)dt + 9.

1 x, u(0) = 5 2

(x − t)u(t)dt +

1 2

(x − t)u(t)dt −

1 24

(x − t)2 u(t)dt −

x

0 x

x 0

sinh(x − t)u (t)dt =

1 x 1 xe − sinh x, 2 2

sinh(x − t)u (t)dt = xex , u(0) = u (0) = u (0) = 1 1 2 x , u(0) = u (0) = 1, u (0) = −1 2

(x − t)4 u (t)dt =

x

0

1 1 x − x cos x, u(0) = 0, u (0) = 1 2 2

(x − t)u (t)dt =

(x − t)2 u (t)dt =

0

1 12

1 3 x + sin x, u(0) = −1, u (0) = 1 6

1 3 x , u(0) = u (0) = 0, u (0) = 2, 3

(x − t)3 u (t)dt =

1 4 x , u(0) = u (0) = 0, u (0) = 3 12

5.3.2 The Variational Iteration Method The standard form of the Volterra integro-diﬀerential equation [11–12] of the ﬁrst kind is given by x x K1 (x, t)u(t)dt + K2 (x, t)u(n) (t)dt = f (x), K2 (x, t) = 0, (5.206) 0

0

where initial conditions are prescribed. We ﬁrst diﬀerentiate both sides of (5.206) to convert it to an equivalent Volterra integro-diﬀerential equation of the second kind, where Leibnitz rule should be used for diﬀerentiating the integrals at the left side, hence we obtain

5.3 Volterra Integro-Diﬀerential Equations of the First Kind

207

x ∂(K1 (x, t)) (n) K1 (x, x) 1 f (x) − u(x) − u (t)dt K2 (x, x) K2 (x, x) K2 (x, x) 0 ∂x x ∂(K2 (x, t)) (n) 1 u (t)dt, K2 (x, x) = 0. − K2 (x, x) 0 ∂x (5.207) As stated before, to use the variational iteration method, we should ﬁrst determine the Lagrange multiplier λ. The Lagrange multiplier λ can be determined based on the resulting integro-diﬀerential equations, where the following rule for λ 1 u(n) + f (u(t), u (t), u (t), · · · , u(n) (t)) = 0, λ = (−1)n (t − x)(n−1) , (n − 1)! (5.208) was derived. Having λ determined, an iteration formula, should be used for the determination of the successive approximations un+1 (x), n 0 of the solution u(x). The zeroth approximation u0 can be any selective function. However, using the initial values u(0), u (0), . . . are preferably used for the selective zeroth approximation u0 as will be seen later. Consequently, the solution is given by (5.209) u(x) = lim un(x). u(n) (x) =

n→∞

The VIM will be illustrated by studying the following examples. Example 5.29 Solve the Volterra integro-diﬀerential equation of the ﬁrst kind x 1 (x − t + 1)u (t)dt = ex + x2 − 1, u(0) = 1. (5.210) 2 0 Diﬀerentiating both sides of (5.210) once with respect to x gives the Volterra integro-diﬀerential equation of the second kind x u (x) = ex + x − u (t)dt. (5.211) 0

The correction functional for Eq. (5.211) is given by x t un (r)dr dt. un (t) − et − t + un+1 (x) = un (x) − 0

(5.212)

0

where we used λ(t) = −1. The zeroth approximation u0 (x) can be selected by u0 (x) = 1. This gives the successive approximations u0 (x) = 1, 1 2 x , 2! 1 1 u2 (x) = 1 + x + x2 − x3 , 2! 3! 1 1 u3 (x) = ex − x3 + x4 , 3! 4! u1 (x) = ex +

(5.213)

208

5 Volterra Integro-Diﬀerential Equations

1 2 x + 2! 1 u5 (x) = 1 + x + x2 + 2! .. . .

u4 (x) = 1 + x +

This gives

1 4 x − 4! 1 4 x + 4!

1 5 x , 5! 1 6 x 6!

1 1 1 un (x) = x + 1 + x2 + x4 + x6 + · · · . 2! 4! 6!

(5.214)

The exact solution is therefore given by u(x) = x + cosh x.

(5.215)

Example 5.30 Use the variational iteration method to solve the Volterra integro-diﬀerential equation of the ﬁrst kind x 1 x 1 (x − t)u(t)dt − (x − t + 1)u (t)dt = 2x2 − x, u(0) = 5. (5.216) 2 2 0 0 Diﬀerentiating both sides of (5.216) once with respect to x gives the Volterra integro-diﬀerential equation of the second kind x (2u(t) − u (t))dt. (5.217) u (x) = −8x + 1 + 0

The correction functional for equation this equation is given by x t un (t) + 8t − 1 − un+1 (x) = un (x) − (2un (r) − un (r)) dr dt. 0

0

(5.218) where we used λ(t) = −1. The zeroth approximation u0 (x) can be selected by u0 (x) = 5. This gives the successive approximations u0 (x) = 5, u1 (x) = 5 + x + x2 , 1 1 u2 (x) = 5 + x + x2 + x4 , 2! 3! 1 1 1 1 u3 (x) = 5 + x + x2 + x3 + x4 − x5 + · · · , 2! 3! 12 30 (5.219) 1 2 1 3 1 4 1 6 u4 (x) = 5 + x + x + x + x + x + · · · , 2! 3! 4! 90 1 2 1 3 1 4 1 u5 (x) = 5 + x + x + x + x + x5 + · · · , 2! 3! 4! 5! .. . . This gives 1 2 1 3 1 4 1 5 (5.220) un (x) = 4 + 1 + x + x + x + x + x + · · · . 2! 3! 4! 5!

5.3 Volterra Integro-Diﬀerential Equations of the First Kind

209

The exact solution is therefore given by u(x) = 4 + ex .

(5.221)

Example 5.31 Use the variational iteration method to solve the Volterra integro-diﬀerential equation of the ﬁrst kind x x 1 (x − t)u(t)dt + (x − t + 1)u (t)dt = sin x + x3 , u(0) = −1, u (0) = 1. 3! 0 0 (5.222) Diﬀerentiating both sides of this equation once with respect to x gives the Volterra integro-diﬀerential equation of the second kind x x 1 u (x) = cos x + x2 − u(t)dt − u (t)dt. (5.223) 2 0 0 The correction functional for equation this equation is given by x un+1 (x) = un (x) + ((t − x)Γ(t)) dt. (5.224) 0

where

t 1 (un (r) + un (r)) dr . un (t) − cos t − t2 + 2 0 The zeroth approximation u0 (x) can be selected by u0 (x) = −1 + x the initial conditions. This gives the successive approximations u0 (x) = −1 + x, 1 u1 (x) = x + x3 − cos x, 3! 1 1 u2 (x) = −1 + x + x2 − x4 + · · · , 2! 12 1 1 u3 (x) = −1 + x + x2 − x4 + · · · , 2! 4! 1 1 u4 (x) = −1 + x + x2 − x4 + · · · , 2! 4! 1 2 1 1 1 u5 (x) = −1 + x + x − x4 + x6 − x8 + · · · , 2! 4! 6! 8! .. . . This gives 1 2 1 4 1 6 1 8 un (x) = x − 1 − x + x − x + x + · · · . 2! 4! 6! 8! The exact solution is therefore given by Γ(t) =

u(x) = x − cos x.

(5.225) by using

(5.226)

(5.227)

(5.228)

210

5 Volterra Integro-Diﬀerential Equations

Example 5.32 Use the variational iteration method to solve the Volterra integro-diﬀerential equation of the ﬁrst kind x x sinh(x − t)u(t)dt + cosh(x − t)u (t)dt = xex , (5.229) 0 0 u(0) = u (0) = u (0) = 1. Diﬀerentiating both sides gives the Volterra integro-diﬀerential equation of the second kind u (x) = xex + ex −

x

0

x

cosh(x − t)u(t)dt −

0

sinh(x − t)u (t)dt. (5.230)

The correction functional for equation this equation is given by

1 x (t − x)2 Γ1 (t) dt, un+1 (x) = un (x) − (5.231) 2 0 where t t t t (t) − te − e + cosh(t − r)u (r)dr + u (r) dr . Γ1 (t) = u n n n 0

0

(5.232) The zeroth approximation can be selected by u0 (x) = 1 + x + 12 x2 by using the initial conditions. This gives the successive approximations 1 u0 (x) = 1 + x + x2 , 2! 1 1 1 1 u1 (x) = 1 + x + x2 + x3 + x4 + x5 + · · · , 2! 3! 4! 60 (5.233) 1 2 1 3 1 4 1 5 1 6 u2 (x) = 1 + x + x + x + x + x + x + · · · , 2! 3! 4! 5! 6! .. . . This gives 1 1 1 1 1 (5.234) un (x) = 1 + x + x2 + x3 + x4 + x5 + x6 + · · · . 2! 3! 4! 5! 6! The exact solution is therefore given by u(x) = ex .

(5.235)

Exercises 5.3.2 Solve the following Volterra integro-diﬀerential equations of the ﬁrst kind by using the variational iteration method: x 3 1 x 3 1. (x − t)u(t)dt − (x − t + 1)u (t)dt = + x − ex , u(0) = 0 2 2 2 0 0 x x (x − t)u(t)dt + (x − t + 1)u (t)dt = 1 + x − cos x, u(0) = 0 2. 0

0

References x 3. (x − t + 1)u (t)dt = 1 + sin x − cos x, u(0) = 0

0 x

4.

0 x

5. 0 x

6.

0 x

7. 0

8.

(x − t)u(t)dt +

x

0

(x − t + 1)u (t)dt =

211

1 2 1 3 x + x + sin x, u(0) = 1 2 6

(x − t + 1)u (t)dt = 2 sin x − x, u(0) = −1, u (0) = 1 (x − t)u(t)dt −

x

0

(x − t + 1)u (t)dt =

sinh(x − t)u(t)dt +

x

0

1 3 x − sinh x, u(0) = 1, u (0) = 1 6

cosh(x − t)u (t)dt =

1 x 1 xe + sinh x, 2 2

u(0) = 1, u (0) = −1 x x (x − t)u(t)dt + (x − t + 2)u (t)dt = 2x, u(0) = 0, u (0) = 1

0 x

9. 0

0

(x − t − 1)u(t)dt +

x

0

(x − t + 1)u (t)dt = 2x − 4 sin x,

u(0) = u (0) = 1, u (0) = −1 x x 10. (x − t + 1)u(t)dt − (x − t − 1)u (t)dt = 2x − 4 sin x, 0

0

u(0) = −1, u (0) = u (0) = 1 x x 11. sinh(x − t)u(t)dt − cosh(x − t)u (t)dt = −x + sinh x, 0

0

u(0) = u (0) = u (0) = 1 x 1 1 x 1 12. (x − t)u(t)dt − (x − t + 1)u (t)dt = − x + x3 , 2 2 6 0 0 u(0) = u (0) = 1, u (0) = 2

References 1. L.G. Chambers, Integral Equations, A Short Course, International Textbook Company, London, (1976). 2. R.F. Churchhouse, Handbook of Applicable Mathematics, Wiley, New York, (1981). 3. B.L. Moiseiwitsch, Integral Equations, Longman, London and New York, (1977). 4. V. Volterra, Theory of Functionals of Integral and Integro-Diﬀerential Equations, Dover, New York, (1959). 5. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, San Diego, (1986). 6. G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994). 7. G. Adomian and R. Rach, Noise terms in decomposition series solution, Comput. Math. Appl., 24 (1992) 61–64. 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 9. A.M.Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997).

212

5 Volterra Integro-Diﬀerential Equations

10. A.M. Wazwaz, The variational iteration method; a reliable tool for solving linear and nonlinear wave equations, Comput. Math. Appl., 54 (2007) 926–932. 11. P. Linz, A simple approximation method for solving Volterra integro-diﬀerential equations of the ﬁrst kind, J. Inst. Math. Appl., 14 (1974) 211–215. 12. P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, (1985).

Chapter 6

Fredholm Integro-Diﬀerential Equations

6.1 Introduction In Chapter 2, the conversion of boundary value problems to Fredholm integral equations was presented. However, the research work in this ﬁeld resulted in a new speciﬁc topic, where both diﬀerential and integral operators appeared together in the same equation. This new type of equations, with constant limits of integration, was termed as Fredholm integro-diﬀerential equations, given in the form b u(n) (x) = f (x) + K(x, t)u(t)dt, u(k) (0) = bk , 0 k n − 1, (6.1) a n

where u(n) (x) = ddxnu . Because the resulted equation in (6.1) combines the diﬀerential operator and the integral operator, then it is necessary to deﬁne initial conditions u(0), u (0), . . . , u(n−1) (0) for the determination of the particular solution u(x) of equation (6.1). Any Fredholm integro-diﬀerential equation is characterized by the existence of one or more of the derivatives u (x), u (x),. . . outside the integral sign. The Fredholm integro-diﬀerential equations of the second kind appear in a variety of scientiﬁc applications such as the theory of signal processing and neural networks [1–3]. A variety of numerical and analytical methods are used in the literature to solve the Fredholm integro-diﬀerential equations. In this chapter, these equations will be handled by using the traditional methods, namely, the direct computation method and the Taylor series method. Moreover, these equations will be handled by the newly developed methods, namely the variational iteration method, and the Adomian decomposition method.

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

214

6 Fredholm Integro-Diﬀerential Equations

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind In this chapter, we will focus our study on the equations that involve separable kernels where the kernel K(x, t) can be expressed as a ﬁnite sum of the form n gk (x) hk (t). (6.2) K(x, t) = k=1

Without loss of generality, we will make our analysis on a one term kernel K(x, t) of the form K(x, t) = g(x) h(t). (6.3) Other cases can be examined in a like manner. The non-separable kernel can be reduced to separable kernel by using the Taylor expansion for the kernel involved. However, the separable kernels only will be presented in this text. The methods that will be used were presented before in details, but here we will outline the main steps of each method to illustrate its use.

6.2.1 The Direct Computation Method The direct computation method has been introduced in Chapter 4. The standard form of the Fredholm integro-diﬀerential equation is given by b (n) K(x, t)u(t)dt, u(k) (0) = bk , 0 k n − 1 (6.4) u (x) = f (x) + a

where u(n) (x) indicates the nth derivative of u(x) with respect to x and bk are the initial conditions. Substituting (6.3) into (6.4) gives b (n) h(t)u(t)dt, u(k) (0) = bk , 0 k n − 1. (6.5) u (x) = f (x) + g(x) a

We can easily observe that the deﬁnite integral in the integro-diﬀerential equation (6.5) involves an integrand that depends completely on the variable t. This means that the deﬁnite integral at the right side of (6.5) is equivalent to a constant α. In other words, we set b α= h(t)u(t)dt. (6.6) a

Consequently, Equation. (6.5) becomes (6.7) u(n) (x) = f (x) + α g(x). Integrating both sides of (6.7) n times from 0 to x, and using the prescribed initial conditions, we can ﬁnd an expression for u(x) that involves the constant α in addition to the variable x. This means we can write u(x) = v(x; α).

(6.8)

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

215

Substituting (6.8) into the right side of (6.6), evaluating the integral, and solving the resulting equation, we determine a numerical value for the constant α. This leads to the exact solution u(x) obtained upon substituting the resulting value of α into (6.8). It is important to recall that this method leads always to the exact solution and not to series components. The method was used before in Chapter 4 for handling Fredholm integral equations. The method will be illustrated by studying the following examples. Example 6.1 Solve the following Fredholm integro-diﬀerential equation 1 u (x) = 3 + 6x + x tu(t)dt, u(0) = 0.

(6.9)

0

This equation may be written as u (x) = 3 + 6x + αx, obtained by setting

u(0) = 0,

(6.10)

1

α=

tu(t)dt.

(6.11)

0

Integrating both sides of (6.10) from 0 to x, and by using the given initial condition we obtain 1 (6.12) u(x) = 3x + 3x2 + αx2 . 2 Substituting (6.12) into (6.11) and evaluating the integral yield 1 7 1 α= (6.13) tu(t)dt = + α, 4 8 0 hence we ﬁnd α = 2. (6.14) The exact solution is given by u(x) = 3x + 4x2 .

(6.15)

Example 6.2 Solve the following Fredholm integro-diﬀerential equation 1 u(t)dt, u(0) = u (0) = 1. u (x) = 1 − e + ex +

(6.16)

0

This equation may be written as u (x) = 1 − e + ex + α, u(0) = 1, obtained by setting 1 α= u(t)dt.

u (0) = 1,

(6.17) (6.18)

0

Integrating both sides of (6.17) twice from 0 to x, and by using the given initial conditions we obtain

216

6 Fredholm Integro-Diﬀerential Equations

u(x) − x − 1 =

1−e+α 2 x + ex − 1 − x, 2

(6.19)

or equivalently

1−e+α 2 (6.20) x + ex . 2 Substituting (6.20) into (6.18) and evaluating the integral we obtain 1 1−e+α u(t)dt = α= + e − 1, (6.21) 6 0 hence we ﬁnd α = e − 1. (6.22) u(x) =

The exact solution is given by u(x) = ex .

(6.23)

Example 6.3 Solve the following Fredholm integro-diﬀerential equation π u (x) = 2+sin x− (x−t)u(t)dt, u(0) = 1, u (0) = 0, u (0) = −1. (6.24) 0

This equation may be written as u (x) = 2 + β − αx + sin x, u(0) = 1, u (0) = 0, u (0) = −1, obtained by setting π π u(t)dt, β = tu(t)dt. α= 0

(6.25)

(6.26)

0

Integrating both sides of (6.25) three times from 0 to x, and by using the given initial conditions we obtain 1 α u(x) = (2 + β)x3 − x4 + cos x. (6.27) 3! 4! Substituting (6.27) into (6.26) and evaluating the integrals we ﬁnd α α 6 π5 π4 (2 + β) − π5 , β = (2 + β) − π − 2. (6.28) α= 4! 5! 30 144 Solving this system of equations gives α = 0, β = −2. (6.29) The exact solution is given by u(x) = cos x.

(6.30)

Example 6.4 Solve the following Fredholm integro-diﬀerential equation π2 (iv) (x − 2t)u(t)dt, u (x) = (2x − π) + sin x + cos x − 0

u(0) = u (0) = 1, u (0) = u (0) = −1. This equation may be written as

(6.31)

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

u(iv) (x) = (2 − α)x + (2β − π) + sin x + cos x, u(0) = u (0) = 1, u (0) = u (0) = −1, obtained by setting π2 π2 α= u(t)dt, β = tu(t)dt. 0

217

(6.32)

(6.33)

0

Integrating both sides of (6.32) four times from 0 to x, and by using the given initial conditions we obtain 1 π 1 β − α x5 + − x4 . u(x) = sin x + cos x + (6.34) 60 120 12 24 Substituting (6.34) into (6.33) and evaluating the integrals we ﬁnd π5 π6 (10 + α) + (2 + β), α=− 46080 1920 (6.35) π7 π π6 β=− (29 + 3α) + ( + β). 322560 2 4608 Solving this system of equations gives π α = 2, β = . (6.36) 2 The exact solution is therefore given by u(x) = sin x + cos x, (6.37) obtained upon substituting (6.36) into (6.34).

Exercises 6.2.1 Solve the following Fredholm integro-diﬀerential equations by using the direct computation method 1 1. u (x) = 12x + u(t)dt, u(0) = 0 0

2. u (x) = 36x2 +

1

0

u(t)dt, u(0) = 1

3. u (x) = sec2 x − ln 2 + 4. u (x) = −10x +

1 −1

π 3

0

(x − t)u(t)dt, u(0) = 1

5. u (x) = 2 sec2 x tan x − 1 + 6. u (x) = −1 + cos x +

π 2

0

7. u (x) = −1 − sin x + 8. u (x) = 2 − cos x +

u(t)dt, u(0) = 0

0 π

0

π 2

π/4

0

u(t)dt, u(0) = 1

tu(t)dt, u(0) = 0 tu(t)dt, u(0) = 0, u (0) = 1

tu(t)dt, u(0) = 1, u (0) = 0

218

6 Fredholm Integro-Diﬀerential Equations

9. u (x) = −2x − sin x + cos x +

10. u (x) = 1 − 3 ln 2 + 4 cosh x +

π

11. u (x) = 2x − x sin x + 2 cos x + 12. u (x) = 4x − sin x +

xu(t)dt, u(0) = −1, u (0) = 1

0

ln 2

0

π 2

−π

(t − x)u(t)dt, u(0) = 0, u (0) = 0

2

π 2

−π

(x − t)2 u(t)dt, u(0) = 0, u (0) = 1

2

13. u (x) = 5 ln 2 − 3 − x + 4 cosh x + u(0) = u (0) = 0, u (0) = 4 14. u (x) = e − 2 − x + ex (3 + x) +

tu(t)dt, u(0) = 4, u (0) = 0

ln 2

0

1

0

(x − t)u(t)dt,

(x − t)u(t)dt,

u(0) = 0, u (0) = 1, u (0) = 2 π 2 tu(t)dt, u(0) = u (0) = 0, u (0) = 1, u (0) = −1 15. u(iv) (x) = −1 + sin x + (iv)

16. u

(x) = 4x + sin x +

0

π 2

−π

(x − t)2 u(t)dt, u(0) = u (0) = 0,

2

u (0) = 1, u (0) = −1

6.2.2 The Variational Iteration Method The variational iteration method [4] was used in Chapters 3, 4 and 5. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists. The standard ith order Fredholm integro-diﬀerential equation is of the form b

u(i) (x) = f (x) +

K(x, t)u(t)dt,

(6.38)

a i

where u(i) (x) = ddxui , and u(0), u (0), . . . , u(i−1) (0) are the initial conditions. The correction functional for the integro-diﬀerential equation (6.38) is x b un+1 (x) = un (x) + λ(ξ) u(i) K(ξ, r)˜ un (r)dr dξ. (6.39) n (ξ) − f (ξ) − 0

a

As presented before, the variational iteration method is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ(ξ) that can be identiﬁed optimally via integration by parts and by using a restricted variation. Having λ(ξ) determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1(x), n 0 of the solution u(x). The zeroth approxima-

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

219

tion u0 can be any selective function. However, using the given initial values u(0), u (0), . . . are preferably used for the selective zeroth approximation u0 as will be seen later. Consequently, the solution is given by u(x) = lim un (x). (6.40) n→∞

The VIM will be illustrated by studying the following examples. Example 6.5 Use the variational iteration method to solve the Fredholm integro-diﬀerential equation π2 tu(t)dt, u(0) = 0. (6.41) u (x) = −1 + cos x + 0

The correction functional for this equation is given by x π2 run (r) dr dξ, un+1 (x) = un (x) − un (ξ) + 1 − cos ξ −

(6.42)

0

0

where we used λ = −1 for ﬁrst-order integro-diﬀerential equations. Notice that the correction functional is the so-called Volterra-Fredholm integral equation, because it involves both types of integral equations. We can use the initial condition to select u0 (x) = u(0) = 0. Using this selection into the correction functional gives the following successive approximations u0 (x) = 0, x π2 u1 (x) = u0 (x) − ru0 (r)dr dξ u0 (ξ) + 1 − cos ξ − 0

0

= sin x − x, x u1 (ξ) + 1 − cos ξ − u2 (x) = u1 (x) − 0

0

π3 x , = (sin x − x) + x − 24 x u2 (ξ) + 1 − cos ξ − u3 (x) = u2 (x) − 0

π 2

π 2

0

ru1 (r)dr

dξ

ru2 (r)dr

(6.43) dξ

3 π π6 π3 x + x− x , = (sin x − x) + x − 24 24 576 x π2 u4 (x) = u3 (x) − ru3 (r)dr dξ u3 (ξ) + 1 − cos ξ −

0

0

3 6 π π6 π π3 x + x− x + x + ··· , = (sin x − x) + x − 24 24 576 576 and so on. The VIM admits the use of u(x) = lim un (x). n→∞

(6.44)

220

6 Fredholm Integro-Diﬀerential Equations

It is obvious that noise terms appear in the successive approximations, where by canceling these noise terms, we obtain the exact solution u(x) = sin x. (6.45) Example 6.6 Use the variational iteration method to solve the Fredholm integro-diﬀerential equation π u (x) = 2 − cos x + tu(t)dt, u(0) = 1, u (0) = 0. (6.46) 0

The correction functional for this equation is given by x π un+1 (x) = un (x)+ (ξ−x) un (ξ) − 2 + cos ξ − run (r) dr dξ, (6.47) 0

0

where λ = ξ − x for second-order integro-diﬀerential equations. We can use the initial conditions u(0) = 1, u (0) = 0 to select the zeroth approximation by u0 (x) = u(0) + xu (0) = 1. (6.48) Using this selection into the correction functional gives the following successive approximations u0 (x) = 1, x π u1 (x) = u0 (x) + (ξ − x) u0 (ξ) − 2 + cos ξ − ru0 (r)dr dξ, 0

0

π2 2 , = cos x + x 1 + 4 x (ξ − x) u1 (ξ) − 2 + cos ξ − u2 (x) = u1 (x) + 0

0

0

0

π

ru1 (r)dr dξ,

4 (6.49) π2 π2 π π6 2 2 2 −x 1+ +x + , = cos x + x 1 + 4 4 8 32 x π u3 (x) = u2 (x) + (ξ − x) u2 (ξ) − 2 + cos ξ − ru2 (r)dr dξ, 4 π2 π2 π π6 2 2 2 −x 1+ +x + = cos x + x 1 + 4 4 8 32 4 6 π π + + ··· , −x2 8 32 and so on. The VIM admits the use of u(x) = lim un (x). n→∞

Canceling the noise terms gives the exact solution u(x) = cos x.

(6.50) (6.51)

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

221

Example 6.7 Use the variational iteration method to solve the second-order Fredholm integro-diﬀerential equation π2 u (x) = 2x − x sin x + 2 cos x − (x − t)u(t)dt, u(0) = u (0) = 0. (6.52) −π 2

The correction functional for this equation is given by x un+1 (x) = un (x) + (ξ − x)Γn (ξ)dξ, n 0,

(6.53)

0

where Γn (ξ) = un (ξ) − 2ξ + ξ sin ξ − 2 cos ξ +

π 2

−π 2

(ξ − r)un (r) dr.

(6.54)

We can use the initial conditions u(0) = 0, u (0) = 0 to select the zeroth approximation by (6.55) u0 (x) = u(0) + xu (0) = 0. Using this selection into the correction functional gives the following successive approximations u0 (x) = 0, x 1 (ξ − x)Γ0 (ξ)dξ = x sin x + x3 , u1 (x) = u0 (x) + 3 0 x u2 (x) = u1 (x) + (ξ − x)Γ1 (ξ)dξ 0

π5 2 1 3 1 3 x , = x sin x + x + − x + 3 3 480 x u3 (x) = u2 (x) + (ξ − x)Γ2 (ξ)dξ

(6.56)

0

π5 2 1 3 π5 2 1 3 x + − x + ··· , = x sin x + x + − x + 3 3 480 480 and so on,where π2 Γi (ξ) = ui (ξ) − 2ξ + ξ sin ξ − 2 cos ξ + (ξ − r)ui (r) dr, i 0. (6.57) −π 2

Canceling the noise terms gives the exact solution u(x) = x sin x.

(6.58)

Example 6.8 Use the variational iteration method to solve the third-order Fredholm integro-diﬀerential equation 1 tu(t)dt, u(0) = 1, u (0) = 1, u (0) = 1. (6.59) u (x) = ex − 1 + 0

222

6 Fredholm Integro-Diﬀerential Equations

The correction functional for this equation is given by 1 1 x ξ un+1 (x) = un (x) − (ξ − x)2 u (ξ) − e + 1 − ru (r) dr dξ, n n 2 0 0 (6.60) where λ = − 12 (ξ − x)2 for third-order integro-diﬀerential equations. The zeroth approximation u0 (x) can be selected by 1 1 u0 (x) = u(0) + xu (0) + x2 u (0) = 1 + x + x2 . (6.61) 2! 2! Using this selection into the correction functional gives the following successive approximations 1 u0 (x) = 1 + x + x2 , 2! 1 1 x 2 ξ (ξ − x) u0 (ξ) − e + 1 − ru0 (r)dr dξ u1 (x) = u0 (x) − 2 0 0 1 3 x , = ex − 144 1 1 x 2 ξ (ξ − x) u1 (ξ) − e + 1 − ru1 (r)dr dξ (6.62) u2 (x) = u1 (x) − 2 0 0 1 3 29 3 1 3 x + x − x , = ex − 144 144 4320 1 1 x 2 ξ (ξ − x) u2 (ξ) − e + 1 − ru2 (r)dr dξ u3 (x) = u2 (x) − 2 0 0 1 3 29 3 1 3 29 3 x x + x − x + x + ··· , = e − 144 144 4320 4320 and so on. Canceling the noise terms gives the exact solution u(x) = ex .

(6.63)

Exercises 6.2.2 Solve the following Fredholm integro-diﬀerential equations by using the variational iteration method 1 u(t)dt, u(0) = 0 1. u (x) = 12x + 0

2. u (x) = −4 + 6x +

1

0

3. u (x) = sec2 x − ln 2 + 4. u (x) = −8x +

1

−1

u(t)dt, u(0) = 2

π 3

0

u(t)dt, u(0) = 0

(x − t)u(t)dt, u(0) = 2

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

5. u (x) = 2 sec2 x tan x − 1 + 6. u (x) = −1 + cos x +

7. u (x) = −1 − sin x + 8. u (x) = −2 − cos x +

π 2

π 2

u(t)dt, u(0) = 1

0

tu(t)dt, u(0) = 0, u (0) = 1

0

π/4

(2 − t)u(t)dt, u(0) = 0

0

π

(1 − t)u(t)dt, u(0) = 1, u (0) = 0

0

9. u (x) = −2x − sin x + cos x +

π

0

10. u (x) = −4 + 3 ln 2 + 4 cosh x + 11. u (x) = 2x − x sin x + 2 cos x + 12. u (x) = 4π − cos x +

223

xu(t)dt, u(0) = −1, u (0) = 1

ln 2 0 π 2

−π

(1 − t)u(t)dt, u(0) = 4, u (0) = 0

(t − x)u(t)dt, u(0) = 0, u (0) = 0

2

π

−π

(x − t)2 u(t)dt, u(0) = 1, u (0) = 0

13. u (x) = 5 ln 2 − 3 − x + 4 cosh x +

ln 2

0

(x − t)u(t)dt,

u(0) = u (0) = 0, u (0) = 4 1 tu(t)dt, u(0) = u (0) = u (0) = 1 14. u (x) = −1 + ex + 0

(iv)

15. u

(x) = −1 + sin x +

16. u(iv) (x) = 2 + cos x +

0 π

0

π 2

tu(t)dt, u(0) = u (0) = 0, u (0) = 1, u (0) = −1

tu(t)dt, u(0) = 1, u (0) = u (0) = 0, u (0) = −1

6.2.3 The Adomian Decomposition Method The Adomian decomposition method was presented before and used thoroughly in previous chapters. This method was used in its standard form to obtain the solution in an inﬁnite series of components that can be recurrently determined. In addition, it was used jointly with the noise terms phenomenon. Moreover, a modiﬁed form was developed to facilitate the computational work. The obtained series may give the exact solution if such a solution exists. Otherwise, the series gives an approximation for the solution that gives high accuracy level. The main focus in this section will be directed to converting a Fredholm integro-diﬀerential equation to an equivalent Fredholm integral equation. The converted Fredholm integral equation can then be solved by any of the methods presented before in Chapter 4, such as the direct computation method, successive approximations method, and others. However, in this section we

224

6 Fredholm Integro-Diﬀerential Equations

will use the Adomian decomposition method for solving the Fredholm integrodiﬀerential equation by converting it ﬁrst to an integral equation. The Adomian decomposition method was presented in details in previous chapters, only a summary will be outlined here. Without loss of generality, we may assume a second order Fredholm integro-diﬀerential equation given by b u (x) = f (x) + K(x, t)u(t)dt, u(0) = a0 , u (0) = a1 . (6.64) a

Integrating both sides of (6.64) from 0 to x twice gives a −1 −1 K(x, t) u(t)dt , u(x) = a0 + a1 x + L (f (x)) + L

(6.65)

a

where the initial conditions u(0) = a0 and u (0) = a1 are used, and L−1 is a two-fold integral operator. The Adomian decomposition method admits the use of the decomposition series ∞ u(x) = un (x), (6.66) n=0

into both sides of (6.65) to obtain ∞ un (x) = a0 + a1 x + L−1 (f (x)) + L−1 n=0

x

K(x, t) 0

∞

un (t)dt , (6.67)

n=0

or equivalently u0 (x) + u1 (x) + u2 (x) + u3 (x) + · · · = a0 + a1 x + L−1 (f (x)) b b −1 −1 K(x, t)u0 (t)dt + L K(x, t)u1 (t)dt +L a

+L−1

a

b

K(x, t)u2 (t)dt

(6.68)

+ ··· .

a

Consequently, to determine the components u0 (x), u1 (x), u2 (x), u3 (x), . . . of the solution u(x), we set the recurrence relation u0 (x) = a0 + a1 x + L−1 (f (x)) , b (6.69) uk+1 (x) = L−1 K(x, t)uk (t)dt , k 0, a

where the zeroth component u0 (x) is deﬁned by all terms not included inside the integral sign of (6.68). Having determined the components ui (x), i 0, the solution u(x) of (6.64) is then obtained in a series form. Using (6.66), the obtained series converges to the exact solution if such a solution exists. We point out that some modiﬁed forms of the Adomian method was developed in the literature. The most commonly used form is that one developed in [5–6] and was employed in Chapter 3. In what follows, we summarize the main steps for this modiﬁed form.

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

225

The Modiﬁed Decomposition Method The Adomian decomposition method provides the solutions in an inﬁnite series of components. The method substitutes the decomposition series of u(x), given by ∞ un (x), (6.70) u(x) = n=0

into both sides of the Fredholm integral equation b u(x) = f (x) + λ K(x, t)u(t)dt.

(6.71)

a

The standard Adomian decomposition method introduces the recurrence relation u0 (x) = f (x), b (6.72) K(x, t)uk (t)dt, k 0. uk+1 (x) = λ a

In view of (6.72), the components un (x), n 0 are readily obtained. The modiﬁed decomposition method presents a slight variation to the recurrence relation (6.72) to determine the components of u(x) in an easier and faster manner. For many cases, the function f (x) can be set as the sum of two partial functions, namely f1 (x) and f2 (x). In other words, we can set (6.73) f (x) = f1 (x) + f2 (x). In view of (6.73), we introduce a qualitative change in the formation of the recurrence relation (6.72). The modiﬁed decomposition method identiﬁes the zeroth component u0 (x) by one part of f (x), namely f1 (x) or f2 (x). The other part of f (x) can be added to the component u1 (x) that exists in the standard recurrence relation. The modiﬁed decomposition method admits the use of the modiﬁed recurrence relation u0 (x) = f1 (x), b K(x, t)u0 (t)dt, u1 (x) = f2 (x) + λ (6.74) a b uk+1 (x) = λ K(x, t)uk (t)dt, k 1. a

It is obvious that the diﬀerence between the standard recurrence relation (6.72) and the modiﬁed recurrence relation (6.74) rests only in the formation of the ﬁrst two components u0 (x) and u1 (x) only. The other components uj , j 2 remain the same in the two recurrence relations. It is interesting to recall that the noise terms phenomenon can be used here. This phenomenon was employed before, but it is useful to summarize the main steps of this phenomenon.

226

6 Fredholm Integro-Diﬀerential Equations

The Noise Terms Phenomenon It was shown before that if the noise terms appear between components of u(x), then the exact solution can be obtained only by considering only the ﬁrst two components u0 (x) and u1 (x). The noise terms are deﬁned as the identical terms, with opposite signs, that may appear between the components of the solution u(x). The conclusion made by [5–8] suggests that if we observe the appearance of identical terms in both components with opposite signs, then by canceling these terms, the remaining non-canceled terms of u0 may in some cases provide the exact solution, that should be justiﬁed through substitution. It was formally proved that other terms in other components will vanish in the limit if the noise terms occurred in u0 (x) and u1 (x). However, if the exact solution was not attainable by using this phenomenon, then we should continue determining other components of u(x) in a standard way. The Adomian decomposition method, the noise terms phenomenon, and the modiﬁed decomposition method for solving Fredholm integro-diﬀerential equations will be illustrated by studying the following examples. The selected equations are of orders 1, 2, 3, and 4. Other equations of higher orders can be treated in a like manner. Example 6.9 Use the Adomian decomposition method to solve the Fredholm integrodiﬀerential equation 1 2 u (x) = 36x + u(t)dt, u(0) = 1. (6.75) 0

Recall that the integral at the right side is equivalent to a constant because it depends only on the variable t with constant limits of integration for t. Integrating both sides of Eq. (6.75) from 0 to x gives 1 u(t)dt , (6.76) u(x) − u(0) = 12x3 + x 0

which gives upon using the initial condition 3 u(x) = 1 + 12x + x

1

u(t)dt .

(6.77)

0

Substituting the series assumption u(x) =

∞

un (x),

(6.78)

n=0

into both sides of (6.77) gives ∞ 3 un (x) = 1 + 12x + x n=0

∞ 1 0 n=0

un(t)dt .

(6.79)

The components uj (x), j 0 of u(x) can be determined by using the recurrence relation

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

u0 (x) = 1 + 12x3 ,

uk+1 (x) = x

1 0

uk (t)dt,

This in turn gives

3

u0 (x) = 1 + 12x , u2 (x) = x

u1 (x) = x

1 0

u1 (t)dt = 2x,

u3 (x) = x

k 0.

227

(6.80)

1

u0 (t)dt = 4x,

0 1 0

u2 (t)dt = x,

.. . The solution in a series form is given by 1 1 3 u(x) = 1 + 12x + 4x 1 + + + · · · , 2 4 which gives the exact solution u(x) = 1 + 8x + 12x3 , obtained upon evaluating the sum of the inﬁnite geometric series.

(6.81)

(6.82)

(6.83)

Example 6.10 Use the Adomian decomposition method to solve the Fredholm integrodiﬀerential equation π u (x) = 2 − cos x + tu(t)dt, u(0) = 1, u (0) = 0. (6.84) 0

Integrating both sides of Eq. (6.84) twice from 0 to x gives π 1 2 2 tu(t)dt , u(x) − u(0) − xu (0) = −1 + cos x + x + x 2 0 which gives upon using the initial conditions π 1 2 2 u(x) = cos x + x + x tu(t)dt . 2 0 Substituting the series assumption ∞ un (x), u(x) =

(6.85)

(6.86)

(6.87)

n=0

into both sides of (6.86) gives ∞ ∞ π 1 un (x) = cos x + x2 + x2 t un (t)dt . 2 0 n=0 n=0 Proceeding as before, we set the recurrence relation π 1 u0 (x) = cos x + x2 , uk+1 (x) = x2 tuk (t)dt, k 0. 2 0 This in turn gives 1 2 π π4 2 2 x . u0 (t)dt = −x2 + u0 (x) = cos x + x , u1 (x) = x 2 8 0

(6.88)

(6.89)

(6.90)

228

6 Fredholm Integro-Diﬀerential Equations

The noise terms ±x2 appear between u0 (x) and u1 (x). Canceling the noise term from u0 (x) gives the exact solution u(x) = cos x, (6.91) that justiﬁes the integro-diﬀerential equation. Example 6.11 Use the Adomian decomposition method to solve the Fredholm integrodiﬀerential equation 1 u (x) = ex − x + xtu(t)dt, u(0) = u (0) = u (0) = 1. (6.92) 0

Integrating both sides of Eq. (6.92) three times from 0 to x, and using the initial conditions we obtain 1 1 3 1 3 x u(x) = e − x + x tu(t)dt . (6.93) 3! 3! 0 Proceeding as before, we set the recurrence relation 1 1 1 tuk (t)dt, k 0. (6.94) u0 (x) = ex − x3 , uk+1 (x) = x3 3! 3! 0 This in turn gives 1 u0 (x) = ex − x3 , 3! 1 1 29 3 u1 (x) = x3 tu0 (t)dt = x , 3! 180 0 1 1 29 3 (6.95) tu1 (t)dt = u2 (x) = x3 x , 3! 5400 0 1 1 29 u3 (x) = x3 x3 , tu2 (t)dt = 3! 162000 0 .. . The solution in a series form is given by 1 29 3 1 1 u(x) = ex − x3 + x 1+ + + ··· . 3! 180 30 900 The inﬁnite geometric series has a1 = 1, r = geometric series is given by 30 1 . S= 1 = 29 1 − 30 Using this result gives the exact solution u(x) = ex

1 3.

(6.96)

The sum of the inﬁnite (6.97)

(6.98)

Example 6.12 Use the Adomian decomposition method to solve the Fredholm integrodiﬀerential equation

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

u(iv) (x) = 1 − x + sin x +

u(0) = u (0) = 0,

0

π 2

(x − t)u(t)dt,

229

(6.99)

u (0) = −u (0) = 1.

Integrating both sides of Eq. (6.99) four times from 0 to x and using the initial conditions we ﬁnd π π 1 4 1 4 2 1 3 2 1 3 u(t)dt − x tu(t)dt. (6.100) u(x) = sin x + x − x + x 3! 4! 4! 3! 0 0 Substituting the series assumption and proceeding as before we obtain the recurrence relation 1 1 u0 (x) = sin x + x3 − x4 , 3! 4! π π2 (6.101) 1 4 2 1 uk+1 = x uk (t)dt − x3 tuk (t)dt, k 0. 4! 3! 0 0 This in turn gives 1 1 1 1 u0 (x) = sin x + x3 − x4 , u1 (x) = − x3 + x4 + · · · , 3! 4! 3! 4! (6.102) .. . 1 3 1 4 x , − 4! x 3!

Canceling the noise terms

from u0 (x) gives the exact solution

u(x) = sin x.

(6.103)

Exercises 6.2.3 Solve the following Fredholm integro-diﬀerential equations by using the Adomian decomposition method π 3 u(t)dt, u(0) = 0 1. u (x) = sec2 x − ln 2 + 2. u (x) = −1 + 24x + 3. u (x) = 6 + 17x +

0

1 0

u(t)dt, u(0) = 0

1 0

xu(t)dt, u(0) = 0

√ 4. u (x) = − 3 + 2 sec2 x tan x + π 5. u (x) = − x + sin 2x + 2 6. u (x) = 1 − x + cos x + 7. u (x) = −

π 2

−π

π 3

0

u(t)dt, u(0) = 1

u(t)dt, u(0) = 0

2

π 2

0

π − 2 cos 2x + 4

(x − t)u(t)dt, u(0) = 0

0

π 2

u(t)dt, u(0) = 1, u (0) = 0

230

6 Fredholm Integro-Diﬀerential Equations

8. u (x) = 2x − cos x +

π

xtu(t)dt, u(0) = 1, u (0) = 0

0

9. u (x) = −2x − sin x + cos x + 10. u (x) = −2 − cos x +

π 0

π

0

xu(t)dt, u(0) = −1, u (0) = 1

(x − t)u(t)dt, u(0) = 1, u (0) = 0

11. u (x) = 3 ln 2 − 1 − 3x + 4 cosh x +

ln 2 0

(x − t)u(t)dt, u(0) = 4, u (0) = 0

1 1 1 + (x − t)u(t)dt, u(0) = 0, u (0) = 1 + (1 − 2 ln 2)x − 4 (1 + x)2 0 1 13. u (x) = 2e−1 − e−x + tu(t)dt, u(0) = u (0) = 1, u (0) = −1 12. u (x) =

−1

14. u (x) = e − 2 − x + (3 + x)ex + 15. u(iv) (x) = −2x + sin x + cos x +

1

0 π 2

−π

(x − t)u(t)dt, u(0) = u (0) = 1, u (0) = 2 xtu(t)dt,

2

u(0) = u (0) = 1, u (0) = u (0) = −1 16. u(iv) (x) =

6 1 + (1 − 2 ln 2)x − + 4 (1 + x)4

1 0

(x − t)u(t)dt,

u(0) = 0, u (0) = 1, u (0) = −1, u (0) = 2

6.2.4 The Series Solution Method The series solution method was used before, where the generic form of Taylor series for an analytic solution u(x) at x = 0 ∞ a n xn (6.104) u(x) = n=0

is used. Following the discussion presented before in Chapters 3 and 4, the series solution method will be used for solving Fredholm integro-diﬀerential equations. We will assume that the solution u(x) of the Fredholm integrodiﬀerential equation b u(k) (x) = f (x) + λ K(x, t)u(t)dt, u(j) (0) = aj , 0 j (k − 1), (6.105) a

is analytic, and therefore possesses a Taylor series of the form given in (6.104), where the coeﬃcients an will be determined recurrently. Substituting (6.104) into both sides of (6.105) gives

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

∞

(k) n

an x

b

= T (f (x)) + λ

K(x, t) a

n=0

∞

231

an t

n

dt,

(6.106)

n=0

or for simplicity we use

(k) 2 a0 + a1 x + a2 x + · · · = T (f (x))+λ

b

K(x, t) a0 + a1 t + a2 t2 + · · · dt,

a

(6.107) where T (f (x)) is the Taylor series for f (x). The integral equation (6.105) will be converted to a traditional integral in (6.106) or (6.107) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. Moreover, the given initial equations should be used in the series assumption (6.104). We ﬁrst integrate the right side of the integral in (6.106) or (6.107), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x in both sides of the resulting equation to obtain a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (6.104). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. It is worth noting that using the series solution method for solving Fredholm integro-diﬀerential equations gives exact solutions if the solution u(x) is a polynomial. However, if the solution is any other elementary function such as sin x, ex , etc, the series method gives the exact solution after rounding few of the coeﬃcients aj , j 0. This will be illustrated by studying the following examples. Example 6.13 Solve the Fredholm integro-diﬀerential equation by using the series solution method 1 (1 − xt)u(t)dt, u(0) = 1. (6.108) u (x) = 4 + 4x + −1

Substituting u(x) by the series

u (x) =

∞

an x

n

,

(6.109)

n=0

into both sides of Eq. (6.108) leads to 1 ∞ ∞ n−1 n (1 − xt) dt. nan x = 4 + 4x + an t n=1

−1

n=0

(6.110)

232

6 Fredholm Integro-Diﬀerential Equations

Notice that a0 = 1 from the given initial conditions. Evaluating the integral at the right side gives a1 + 2a2 x + 3a3 x2 + · · · (6.111) 2 2 2 2 2 2 2 2 = 6 + a2 + a4 + a6 + a8 + (4 − a1 − a3 − a5 − a7 )x. 3 5 7 9 3 5 7 9 Equating the coeﬃcients of like powers of x in both sides of (6.111) gives (6.112) a1 = 6, an = 0, n 2. The exact solution is given by u(x) = 1 + 6x,

(6.113)

where we used a0 = 1 from the initial condition. Example 6.14 Solve the Fredholm integro-diﬀerential equation by using the series solution method 1 2 (x − t)u(t)dt, u(0) = 1. (6.114) u (x) = 7 − 2x + 15x + −1

Substituting the series

u (x) =

∞

an x

n

,

(6.115)

n=0

into both sides of Eq. (6.114) leads to ∞ nan xn−1 = 7 − 2x + 15x2 + n=1

1

−1

(x − t)

∞

an t

n

dt.

(6.116)

n=0

Notice that a0 = 1 from the given initial conditions. Evaluating the integral at the right side gives 2 2 2 2 a1 + 2a2 x + 3a3 x2 + · · · = 7 − a1 − a3 − a5 − a7 3 5 7 9 (6.117) 2 2 2 2 + a2 + a4 + a6 + a8 x + 15x2 . 3 5 7 9 Equating the coeﬃcients of like powers of x in both sides of (6.117) gives a1 = 3, a2 = 0, The exact solution is given by

a3 = 5,

an = 0,

u(x) = 1 + 3x + 5x3 , where we used a0 = 1 from the initial condition.

n 4.

(6.118) (6.119)

Example 6.15 Solve the Fredholm integro-diﬀerential equation by using the series solution method 1 5 (xt2 − x2 t)u(t)dt, u(0) = 1, u (0) = 1. (6.120) u (x) = − 11x + 3 −1

6.2 Fredholm Integro-Diﬀerential Equations of the Second Kind

Substituting the series

u (x) =

∞

233

an x

n

,

(6.121)

n=0

into both sides of Eq. (6.120) leads to 1 ∞ ∞ 5 n−2 2 2 n (xt − x t) n(n − 1)an x = − 11x + an t dt. 3 −1 n=2 n=0

(6.122)

Notice that a0 = a1 = 1 from the given initial conditions. Evaluating the integral at the right side gives 31 2 5 2 2 2 2 2a2 + 6a3 x + 12a4 x + · · · = + − + a2 + a4 + a6 + a8 x 3 7 5 7 9 11 2 2 2 2 − + a3 + a5 + a7 x2 . 3 5 7 9 (6.123) Equating the coeﬃcients of like powers of x in both sides of (6.123) gives 5 5 (6.124) a2 = , a3 = − , an = 0, n 4. 6 3 The exact solution is given by 5 5 u(x) = 1 + x + x2 − x3 , (6.125) 6 3 where we used a0 = a1 = 1 from the initial condition. Example 6.16 Solve the Fredholm integro-diﬀerential equation by using the series solution method π u (x) = 1 − x − cos x + xtu(t)dt, u(0) = 0, u (0) = 1, u (0) = 0. (6.126) 0

Substituting the series

u (x) =

∞

an x

n

,

(6.127)

n=0

into both sides of Eq. (6.126) leads to ∞ n−3 n(n − 1)(n − 2)an x = 1 − x − cos x +

0

n=3

π

xt

∞

an t

n

dt. (6.128)

n=0

Evaluating the integral at the right side and proceeding as before we obtain (−1)n (6.129) a2n+1 = , a2n+2 = 0, n 1. (2n + 1)! Recall that the initial conditions give a0 = 0, a1 = 1, a2 = 0. Combining the initial conditions with the last results for the coeﬃcients we ﬁnd u(x) = sin x.

(6.130)

234

6 Fredholm Integro-Diﬀerential Equations

It is to be noted that we used terms up to O(x12 ) in the series to obtain this result.

Exercises 6.2.4 Use the series solution method to solve the following Fredholm integro-diﬀerential equations: 1 1. u (x) = 4x + (x − t)u(t)dt, u(0) = 2 2. u (x) = 4 +

−1

1 −1

(1 − x2 t2 )u(t)dt, u(0) = −2

3. u (x) = 5 + 4x +

1

−1

(x − t)u(t)dt, u(0) = 0

1 10 2 (x2 − t2 )u(t)dt, u(0) = 0 x + 3 −1 1 92 5. u (x) = (x − t)u(t)dt, u(0) = u (0) = 0 + 5 −1 1 6. u (x) = −18 + (x − t)u(t)dt, u(0) = 3, u (0) = 0

4. u (x) = 10x −

−1

7. u (x) = −18 + 8. u (x) = 32x +

1

−1 1

−1

(1 − xt)u(t)dt, u(0) = 3, u (0) = 0

(1 − xt)u(t)dt, u(0) = 1, u (0) = 0

9. u (x) = −2x + sin x − cos x + 10. u (x) = 2 + sin x +

π 0

11. u (x) = −1 + sin x + 12. u (x) = 1 − e + ex +

π 0

xu(t)dt, u(0) = u (0) = 1, u (0) = −1

tu(t)dt, u(0) = 1, u (0) = 0, u (0) = −1

π 2

0 1 0

u(t)dt, u(0) = 1, u (0) = 0, u (0) = −1 u(t)dt, u(0) = u (0) = u (0) = 1

References 1. H.T. Davis, Introduction to Nonlinear Diﬀerential and Integral Equations, Dover, New York, (1962). 2. R.P. Kanwal, Linear Integral Equations, Birkhauser, Boston, (1997). 3. G. Micula and P. Pavel, Diﬀerential and Integral Equations through Practical Problems and Exercises, Kluwer, Boston, (1992). 4. J.H. He, Variational iteration method for autonomous ordinary diﬀerential systems, Appl. Math. Comput., 114(2/3) (2000) 115–123.

References

235

5. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 6. A.M. Wazwaz, A reliable modiﬁcation of the Adomian decomposition method, Appl. Math. Comput., 102 (1999) 77–86. 7. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, San Diego, (1986). 8. G. Adomian, and R. Rach, Noise terms in decomposition series solution, Comput. Math. Appl., 24 (1992) 61–64.

Chapter 7

Abel’s Integral Equation and Singular Integral Equations

7.1 Introduction Abel’s integral equation occurs in many branches of scientiﬁc ﬁelds [1], such as microscopy, seismology, radio astronomy, electron emission, atomic scattering, radar ranging, plasma diagnostics, X-ray radiography, and optical ﬁber evaluation. Abel’s integral equation is the earliest example of an integral equation [2]. In Chapter 2, Abel’s integral equation was deﬁned as a singular integral equation. Volterra integral equations of the ﬁrst kind h(x) f (x) = λ K(x, t)u(t)dt, (7.1) g(x)

or of the second kind

h(x)

K(x, t)u(t)dt,

u(x) = f (x) + λ

(7.2)

g(x)

are called singular [3–4] if: 1. one of the limits of integration g(x), h(x) or both are inﬁnite, or 2. if the kernel K(x, t) becomes inﬁnite at one or more points at the range of integration. Examples of the ﬁrst type are given by Fourier transform and Laplace transform of the function u(x) ∞ e−ıλx u(x)dx, (7.3) F (λ) = −∞

L{u(x)}(s) =

∞

e−sx u(x)dx.

(7.4)

0

respectively. It is obvious that the range of integration is inﬁnite for the Fourier transform and Laplace transform respectively. Examples of the second type are the Abel’s integral equation, generalized Abel’s integral equation, and weakly singular integral equation are given by A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

238

7 Abel’s Integral Equation and Singular Integral Equations

f (x) =

0 x

f (x) = 0

1 u(t)dt, (x − t)

1 u(t)dt, 0 < α < 1, (x − t)α

and

x

u(x) = f (x) + 0

x

1 u(t)dt, 0 < α < 1, (x − t)α

(7.5) (7.6)

(7.7)

respectively. It is clear that the kernel in each equation becomes inﬁnite at the upper limit t = x. The last three singular integral equations are among the earliest integral equations established by the Norwegian mathematician Niles Abel in 1823. In this chapter we will focus our study on the second style of singular integral equations, namely the equations where the kernel K(x, t) becomes unbounded at one or more points of singularities in its domain of deﬁnition. The equations that will be investigated are Abel’s problem, generalized Abel integral equations and the weakly-singular second-kind Volterra type integral equations. In a manner parallel to the approach used in previous chapters, we will focus our study on the techniques that will guarantee the existence of a unique solution to any singular integral equation. We point out here that singular integral equations are in general very diﬃcult to handle.

7.2 Abel’s Integral Equation Abel in 1823 investigated the motion of a particle that slides down along a smooth unknown curve, in a vertical plane, under the inﬂuence of the gravity. The particle takes the time f (x) to move from the highest point of vertical height x to the lowest point 0 on the curve. The Abel’s problem is derived to ﬁnd the equation of that curve. Abel derived the equation of motion of the sliding particle along a smooth curve by the singular integral equation x 1 √ f (x) = u(t)dt, (7.8) x −t 0 where f (x) is a predetermined data function, and u(x) is the solution that will be determined. It is to be noted that Abel’s integral equation (7.8) is also called Volterra integral equation of the ﬁrst kind. Besides the kernel K(x, t) in Abel’s integral equation (7.8) is 1 , (7.9) K(x, t) = √ x−t where K(x, t) → ∞, as t → x. (7.10)

7.2 Abel’s Integral Equation

239

It is interesting to point out that although Abel’s integral equation is a Volterra integral equation of the ﬁrst kind, but two of the methods used before in Section 3.3, namely the series solution method and the conversion to a second kind Volterra equation, are not applicable here. The series solution cannot be used in this case especially if u(x) is not analytic. Moreover, converting Abel’s integral equation to a second kind Volterra equation is not obtainable because we cannot use Leibnitz rule due to the singularity behavior of the kernel in (7.8).

7.2.1 The Laplace Transform Method Although the Laplace transform method was presented before, but a brief summary will be helpful. In the convolution theorem for the Laplace transform method, it was stated that if the kernel K(x, t) of the integral equation x f (x) = K(x, t)u(t)dt, (7.11) 0

depends on the diﬀerence x−t, then it is called a diﬀerence kernel. The Abel’s integral equation can thus be expressed as x K(x − t)u(t)dt. (7.12) f (x) = 0

Consider two functions f1 (x) and f2 (x) that possess the conditions needed for the existence of Laplace transform for each. Let the Laplace transforms for the functions f1 (x) and f2 (x) be given by L{f1 (x)} = F1 (s), (7.13) L{f2 (x)} = F2 (s). The Laplace convolution product of these two functions is deﬁned by x f1 (x − t)f2 (t)dt, (7.14) (f1 ∗ f2 )(x) =

or (f2 ∗ f1 )(x) =

0

x 0

f2 (x − t)f1 (t)dt.

(7.15)

Recall that (f1 ∗ f2 )(x) = (f2 ∗ f1 )(x).

(7.16)

We can easily show that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by L{(f1 ∗ f2 )(x)} = F1 (s)F2 (s). (7.17) Based on this summary, we will examine Abel’s integral equation where the kernel is a diﬀerence kernel. Recall that we will apply the Laplace transform method and the inverse of the Laplace transform using Table 2 in section 1.5. Taking Laplace transforms of both sides of (7.8) leads to

240

7 Abel’s Integral Equation and Singular Integral Equations 1

L{f (x)} = L{u(x)}L{x− 2 }, or equivalently F (s) = U (s)

(7.18)

√ π Γ(1/2) = U (s) , 1/2 1/2 s s

(7.19)

that gives

s1/2 (7.20) U (s) = √ F (s), π √ where Γ is the gamma function, and Γ( 12 ) = π. In Appendix D, the deﬁnition of the gamma function and some of the relations related to it are given. The last equation (7.20) can be rewritten as 1 s √ (7.21) U (s) = ( πs− 2 F (s)), π which can be rewritten by s (7.22) L{u(x)} = L{y(x)}, π where x 1 (x − t)− 2 f (t)dt. (7.23) y(x) = 0

Using the fact

L{y (x)} = sL{y(x)} − y(0),

(7.24)

into (7.22) we obtain L{u(x)} =

1 L{y (x)}. π

(7.25)

Applying L−1 to both sides of (7.25) gives the formula x f (t) 1 d √ dt, (7.26) u(x) = π dx 0 x−t that will be used for the determination of the solution u(x). Notice that the formula (7.26) will be used for solving Abel’s integral equation, and it is not necessary to use Laplace transform method for each problem. Abel’s problem given by (7.8) can be solved directly by using the formula (7.26) where the unknown function u(x) has been replaced by the given function f (x). For f (x) = xn , n is a positive integer, Table 7.1 can be used for evaluating the integral in Abel’s problem. Table 7.1 For integrals of the form u(x) =

x 0

n

√t dt, n x−t

0. 1

u(x)

√ 2c x

4 3 x2 3

16 5 x2 15

32 7 x2 35

.. .

2n+1 Γ(n + 1)xn+ 2 1 · 3 · 5 · · · (2n + 1)

f (t)

c

t

t2

t3

.. .

tn

7.2 Abel’s Integral Equation

241 n

However, for f (x) = x 2 , n is an odd positive integer, Table 7.2 can be used for evaluating the integral in Abel’s problem. Table 7.2 For integrals of the form u(x) = u(x)

√ 2c x

1 πx 2

f (t)

c

t2

1

3 2 πx 8 3

t2

x 0

n

2 √t dt. x−t

5 πx3 16 5

t2

.. .

Γ( n+2 ) √ n+1 2 πx 2 Γ( n+3 ) 2

.. .

n

t2

More details can be found in Appendix B. Using formula (7.26) to determine the solution of Abel’s problem (7.8) will be illustrated by the following examples. Example 7.1 Solve the following Abel’s integral equation x √ 1 √ u(t)dt. 2π x = x−t 0 √ Substituting f (x) = 2π x in (7.26) gives √ x 1 d 2π t d √ u(x) = (πx) = π. dt = π dx 0 dx x−t

(7.27)

(7.28)

Example 7.2 Solve the following Abel’s integral equation x 1 4 32 √ x = u(t)dt. 3 x −t 0

(7.29)

3

Substituting f (x) = 43 x 2 in (7.26) gives x 4 3 t2 1 d 1 d 2 √3 u(x) = (x ) = x dt = π dx 0 2 dx x−t

(7.30)

Example 7.3 Solve the following Abel’s integral equation x 8 3 16 5 1 √ u(t)dt. x2 + x2 = 3 5 x−t 0 Using (7.26) gives x 8 3 16 5 2 2 1 d 3t + 5 t √ u(x) = dt, π dx 0 x−t 1 d (πx2 + πx3 ) = 2x + 3x2 . = π dx

(7.31)

(7.32)

242

7 Abel’s Integral Equation and Singular Integral Equations

Example 7.4 Find two-terms approximation of the solution of the following Abel’s integral equation x 1 √ sin x = u(t)dt, x ∈ [0, 1]. (7.33) x−t 0 Using (7.26) gives

x 1 3 t − 3! t sin t 1 d √ √ dt = dt, π dx x − t x − t 0 0 16 7 8 5 1 1√ 1 d 4 3 2 2 2 x− x = . x − x = π dx 3 105 π 2 15

u(x) =

1 d π dx

x

(7.34)

Exercises 7.2.1 In Exercises 1–12, solve the following Abel’s integral equations where x ∈ [0, 2] x x 1 1 π √ √ u(t)dt 2. x = u(t)dt 1. π = x−t 2 x−t 0 0 x x √ √ 8 3 1 1 √ √ u(t)dt 4. 2 x + x 2 = u(t)dt 3. 2 x = x − t 3 x −t 0 0 x x 3 π 1 1 √ √ u(t)dt 6. πx2 = u(t)dt 5. (x2 − x) = 2 x−t 8 x−t 0 0 x x 1 4 3 4 3 1 1 √ √ u(t)dt 8. πx + x 2 = u(t)dt 7. x 2 = 3 x−t 2 3 x−t 0 0 x x √ 1 1 1 √ √ u(t)dt 10. x3 = u(t)dt 9. 2 x − πx = 2 x − t x −t 0 0 x x 3 1 1 √ √ u(t)dt 12. 3x2 = u(t)dt 11. 4x 2 − x = x−t x−t 0 0 In Exercises 13–16, ﬁnd two-terms approximation for the solution of the following Abel’s integral equations x x 1 1 13. π sinh x = √ √ u(t)dt 14. π sinh x ln(1 + x) = u(t)dt x−t x−t 0 0 x x 1 1 √ √ u(t)dt 16. π sin x tan x = u(t)dt, x 15. π cosh x ln(1 + x) = x−t x−t 0 0

7.3 The Generalized Abel’s Integral Equation Abel generalized his original problem by introducing the singular integral equation x 1 f (x) = u(t)dt, 0 < α < 1, (7.35) (x − t)α 0

7.3 The Generalized Abel’s Integral Equation

243

known as the Generalized Abel’s integral equation where α are known constants such that 0 < α < 1, f (x) is a predetermined data function, and u(x) is the solution that will be determined. The Abel’s problem discussed above is a special case of the generalized equation where α = 12 . The expression (x − t)−α is called the kernel of the Abel integral equation, or simply Abel’s kernel.

7.3.1 The Laplace Transform Method To determine a formula that will be used for solving the generalized Abel’s integral equation (7.35), we will apply the Laplace transform method in a parallel manner to the approach followed before. Taking Laplace transforms of both sides of (7.35) leads to (7.36) L{f (x)} = L{u(x)}L{x−α }, or equivalently Γ(1 − α) F (s) = U (s) 1−α , (7.37) s that gives s1−α F (s), (7.38) U (s) = Γ(1 − α) where Γ is the gamma function. The last equation (7.38) can be rewritten as s L{u(x)} = L(y(x)), (7.39) Γ(α)Γ(1 − α) where x 1 y(x) = f (t)dt. (7.40) α−1 0 (x − t) Using the facts

L{y (x)} = sL{y(x)} − y(0),

and Γ(α)Γ(1 − α) =

π , sin απ

(7.41) (7.42)

into (7.39) we obtain sin απ L(y (x)). (7.43) π Applying L−1 to both sides of (7.43) gives the formula x f (t) sin απ d u(x) = dt. (7.44) π dx 0 (x − t)1−α Integrating the integral at the right side of (7.44) and diﬀerentiating the result we obtain the more suitable x formula f (t) sin απ f (0) + dt , 0 < α < 1. (7.45) u(x) = 1−α π x1−α 0 (x − t) L{u(x)} =

244

7 Abel’s Integral Equation and Singular Integral Equations

Four remarks can be made here: 1. The kernel is called weakly singular as the singularity may be transformed away by a change of variable [5]. 2. The exponent of the kernel of the generalized Abel’s integral equation is −α, but the exponent of the kernel of the two formulae (7.44) and (7.45) is α − 1. 3. The unknown function in (7.35) has been replaced by f (t) and f (t) in (7.44) and (7.45) respectively. 4. In (7.44), diﬀerentiation is used after integrating the integral at the right side, whereas in (7.45), integration is only required. Example 7.5 Solve the following generalized Abel’s integral equation x 128 114 1 x = 1 u(t)dt. 231 0 (x − t) 4

(7.46)

11

4 Notice that α = 14 , f (x) = 128 231 x . Using (7.44) gives √ x 128 11 t4 1 d 2 3 1 d 231 √ = x2 . u(x) = √ πx 3 dt = 3 2π dx 0 (x − t) 4 2π dx

(7.47)

Example 7.6 Solve the following generalized Abel’s integral equation x 1 πx = 2 u(t)dt. 0 (x − t) 3 Notice that α = 23 , f (x) = πx. Using (7.44) gives √ √ √ x 3 3 2 3 d πt 3 d 9 53 u(x) = dt = = x x3 . 2π dx 0 (x − t) 13 2 dx 10 4

(7.48)

(7.49)

Example 7.7 Solve the following generalized Abel’s integral equation x 1 36 11 x6 = 1 u(t)dt. 55 (x − t) 6 0 36 11 x 6 . Using 55 x 36 11 t6 d 55

Notice that α = 16 , f (x) = u(x) =

1 2π dx

0

(x − t)

5 6

(7.50)

(7.44) gives dt =

1 d (πx2 ) = x. 2π dx

(7.51)

Example 7.8 Solve the following generalized Abel’s integral equation x 512 15 1 4 x = 1 u(t)dt. 1155 0 (x − t) 4 Notice that α = 14 , f (x) =

512 15 4 1155 x .

Using (7.44) gives

(7.52)

7.3 The Generalized Abel’s Integral Equation

1 d u(x) = √ 2π dx

0

x

245

512 15 4 1155 t

3

(x − t) 4

dt = x3 .

(7.53)

7.3.2 The Main Generalized Abel Equation It is useful to introduce a further generalization to Abel’s equation by consid1 ering a generalized singular kernel instead of K(x, t) = √x−t . The generalized kernel will be of the form 1 , 0 < α < 1. (7.54) K(x, t) = [g(x) − g(t)]α The main generalized Abel’s integral equation is given by x 1 f (x) = u(t)dt, 0 < α < 1, (7.55) [g(x) − g(t)]α 0 where g(t) is strictly monotonically increasing and diﬀerentiable function in some interval 0 < t < b, and g (t) = 0 for every t in the interval. The solution u(x) of (7.55) is given by x g (t)f (t) sin απ d u(x) = dt, 0 < α < 1. (7.56) π dx 0 [g(x) − g(t)]1−α To prove this formula, we follow [6–7] and consider the integral x g (y)f (y) dy, 1−α 0 [g(x) − g(y)] and substitute for f (u) from (7.55) to obtain x y u(t)g (y) α 1−α dtdy, 0 0 [g(y) − g(t)] [g(x) − g(y)] where by changing the order of integration we ﬁnd y x g (y) u(t)dt α 1−α dy. 0 0 [g(y) − g(t)] [g(x) − g(y)] We can prove that y π g (y) , dy = β(α, 1 − α) = α 1−α sin απ 0 [g(y) − g(t)] [g(x) − g(y)] where β(α, 1 − α) is the beta function. This means that x x π g (y)f (y) dy = u(t)dt. 1−α sin απ 0 0 [g(x) − g(y)] Diﬀerentiating both sides of (7.58) gives x g (y)f (y) sin απ d dy, 0 < α < 1. u(x) = π dx 0 [g(x) − g(y)]1−α

(7.57)

(7.58)

(7.59)

246

7 Abel’s Integral Equation and Singular Integral Equations

It is interesting to point out that the Abel’s integral equation and the generalized Abel’s integral equation are special case of (7.55) by substituting g(x) = x respectively. The following illustrative examples explain how we can use the formula (7.56) in solving the further generalized Abel’s equation. Example 7.9 Solve the following generalized Abel’s integral equation x 4 3 u(t) 4 (sin x) = 1 dt, 3 0 (sin x − sin t) 4

(7.60)

3

where 0 < x < π2 . Notice that α = 14 , f (x) = 43 (sin x) 4 . Besides, g(x) = sin x is strictly monotonically increasing in 0 < x < π2 , and g (x) = cos x = 0 for every x in 0 < x < π2 . Using (7.56) gives 3 x 4 4 1 d 3 cos t(sin t) u(x) = √ 3 dt dx 2π 0 (sin x − sin t) 4 √ (7.61) 4 d 3 2π sin x = cos x. = √ 4 3 2π dx Example 7.10 Solve the following generalized Abel’s integral equation x 2 3 u(t) √ dt, (7.62) πx = 3 x2 − t2 0 where 0 < x < 2. Notice that α = 12 , f (x) = 23 πx3 . Besides, g(x) = x2 is strictly monotonically increasing in 0 < x < 2, and g (x) = 2x = 0 for every x in 0 < x < 2. Using (7.56) gives x 4 4 πt 1 d 1 d π2 4 √3 u(x) = dt = (7.63) x = πx3 . π dx 0 π dx 4 x2 − t 2 Example 7.11 Solve the following generalized Abel’s integral equation x u(t) √ x2 = dt, (7.64) 2 − t2 x 0 where 0 < x < 2. Notice that α = 12 , f (x) = x2 . Besides, g(x) = x2 is strictly monotonically increasing in 0 < x < 2, and g (x) = 2x = 0 for every x in 0 < x < 2. Using (7.56) gives x 1 d 4 3 2t3 4 1 d √ dt = (7.65) u(x) = x = x2 . 2 2 π dx 0 π dx 3 π x −t Example 7.12 Solve the following generalized Abel’s integral equation

7.4 The Weakly Singular Volterra Equations

6 25 x6 = 25

0

x

247

u(t) (x5

1

− t5 ) 6

dt,

(7.66)

25

6 x 6 . Besides, because g(x) is where 0 < x < 2. Notice that α = 16 , f (x) = 25 strictly monotonically increasing in 0 < x < 2, and g (x) = 5x4 = 0 for every x in 0 < x < 2. Using (7.56) gives x 6 49 t6 1 d 1 d 2 5 5 πx = x4 . u(x) = dt = (7.67) 2π dx 0 (x5 − t5 ) 56 2π dx 5

Exercises 7.3 In Exercises 1–8, solve the following generalized Abel’s integral equations x x 9 4 1 1 432 17 x6 = 1. x 3 = 2. 2 u(t)dt 1 u(t)dt 4 935 0 (x − t) 3 0 (x − t) 6 x x 1 1 3 2 9 5 24 5 x6 = x3 = 4. x 3 + 3. 1 u(t)dt 1 u(t)dt 5 2 10 6 0 (x − t) 0 (x − t) 3 x x 1 1 1 243 11 27 7 5. 6. 2πx 3 − x3 = x3 = 1 u(t)dt 2 u(t)dt 440 14 0 (x − t) 3 0 (x − t) 3 x x 1 1 128 11 25 9 16 7 5 = u(t)dt 8. 7. x4 + x4 = x 1 1 u(t)dt 21 231 36 0 (x − t) 4 0 (x − t) 5 In Exercises 9–16, use the formula (7.56) to solve the main generalized Abel’s integral equations x x 3 3 1 1 10. x2 = 9. 2x 2 = 1 u(t)dt 1 u(t)dt 2 0 (x3 − t3 ) 2 0 (x3 − t3 ) 3 x x 2 3 4 1 1 3 3 = 11. x 3 = (sin x) u(t)dt 12. 1 1 u(t)dt 4 2 0 (x2 − t2 ) 3 0 (sin x − sin t) 3 x x 5 1 1 6 2√ x 13. (ex − 1) 6 = 14. e − 1(2ex + 1) = 1 u(t)dt 1 u(t)dt 5 3 0 (ex − et ) 6 0 (ex − et ) 2 x x √ 1 1 64 11 x2 = 15. 2 sin x = 1 u(t)dt 16. 1 u(t)dt 2 231 2 0 (sin x − sin t) 0 (x − t2 ) 4

7.4 The Weakly Singular Volterra Equations The weakly-singular Volterra-type integral equations of the second kind are given by x β √ u(t)dt, x ∈ [0, T ], (7.68) u(x) = f (x) + x −t 0 and

248

7 Abel’s Integral Equation and Singular Integral Equations

x

β (7.69) α u(t)dt, 0 < α < 1, x ∈ [0, T ], 0 [g(x) − g(t)] where β is a constant. Equation. (7.69) is known as the generalized weakly singular Volterra equation. These equations arise in many mathematical physics and chemistry applications such as stereology, heat conduction, crystal growth and the radiation of heat from a semi-inﬁnite solid. It is also assumed that the function f (x) is suﬃciently smooth so that it guarantees a unique solution to (7.68) and to (7.69). The weakly-singular and the generalized weakly-singular equations (7.68) and (7.69) fall under the category of singular equations with singular kernels 1 K(x, t) = √ , x−t (7.70) 1 , K(x, t) = α [g(x) − g(t)] respectively. Notice that the kernel is called weakly singular as the singularity may be transformed away by a change of variable [5]. The weakly-singular Volterra equations (7.68) and (7.69) were investigated by many analytic and numerical methods. In this text we will use only three methods that were used before for handling Volterra integral equations in Chapter 3. u(x) = f (x) +

7.4.1 The Adomian Decomposition Method The Adomian decomposition method has been discussed extensively in this text. We will focus our study on the generalized weakly singular Volterra equation (7.69), because Eq. (7.68) is a special case of the generalized equation with α = 12 , g(x) = x. In the following we outline a brief framework of the method. To determine the solution u(x) of (7.69) we substitute the decomposition series ∞ un (x), (7.71) u(x) = n=0

into both sides of (7.69) to obtain ∞ x ∞ β un (x) = f (x) + un (t) dt, 0 < α < 1. α 0 [g(x) − g(t)] n=0 n=0

(7.72)

The components u0 (x), u1 (x), u2 (x), · · · are usually determined by using the recurrence relation u0 (x) = f (x) x β u1 (x) = α u0 (t)dt, 0 [g(x) − g(t)]

7.4 The Weakly Singular Volterra Equations

u2 (x) = u3 (x) =

x

0 x 0

β u1 (t)dt, [g(x) − g(t)]α β α u2 (t)dt, [g(x) − g(t)]

249

(7.73)

.. . Having determined the components u0 (x), u1 (x), u2 (x), . . ., the solution u(x) of (7.69) will be determined in the form of a rapid convergent power series [8] by substituting the derived components in (7.71). The determination of the previous components can be obtained by using Appendix B, calculator, or any computer symbolic systems such as Maple or Mathematica. The series solution converges to the exact solution if such a solution exists. For concrete problems, we use as many terms as we need for numerical purposes. The method has been proved to be eﬀective in handling this kind of integral equations.

The Modiﬁed Decomposition Method The modiﬁed decomposition method has been used eﬀectively in this text. Recall that this method splits the source term f (x) into two parts f1 (x) and f2 (x), where the ﬁrst part is f1 (x) assigned to the zeroth component u0 (x) and the other part f2 (x) to the ﬁrst component u1 (x). Based on this decomposition of f (x), we use the modiﬁed recurrence relation as follows: u0 (x) = f1 (x), x β u1 (x) = f2 (x) + α u0 (t)dt, [g(x) − g(t)] 0 x β u2 (x) = α u1 (t)dt, (7.74) [g(x) − g(t)] 0 x β u3 (x) = α u2 (t)dt, [g(x) − g(t)] 0 .. . It was proved before that the modiﬁed method accelerates the convergence of the solution.

The Noise Terms Phenomenon The noise terms that may appear between various components of u(x) are deﬁned as the identical terms with opposite signs. By canceling these noise terms between the components u0 (x) and u1 (x) may give the exact solution that should be justiﬁed through substitution. The noise terms, if appeared

250

7 Abel’s Integral Equation and Singular Integral Equations

between components of u(x), accelerate the convergence of the solution and thus minimize the size of the calculations. The noise terms will be used in the following examples. Example 7.13 Solve the weakly singular second kind Volterra integral equation x 16 5 1 2 2 √ u(x) = x + x − u(t)dt. 15 x−t 0 Using the recurrence relation we set 16 5 u0 (x) = x2 + x 2 , 15 x 2 16 5 t + 15 t 2 16 5 π √ dt = − x 2 − x3 . u1 (x) = − 15 3 x−t 0

(7.75)

(7.76)

5

2 The noise terms ± 16 15 x appear between u0 (x) and u1 (x). By canceling the 16 52 noise term 15 x from u0 (x) and verifying that the non-canceled term in u0 (x) justiﬁes the equation (7.75), the exact solution is therefore given by u(x) = x2 . (7.77)

Moreover, we can easily obtain the exact solution by using the modiﬁed decomposition method. This can be done by splitting the nonhomogeneous 5 2 part f (x) into two parts f1 (x) = x2 and f2 (x) = 16 15 x . Accordingly, we set the modiﬁed recurrence relation u0 (x) = x2 , x (7.78) 16 5 t2 2 √ x − u1 (x) = dt = 0. 15 x−t 0 Example 7.14 Solve the weakly singular second kind Volterra integral equation x 1 32 74 4 34 u(x) = 1 − 2x − x + x − (7.79) 1 u(t)dt. 21 3 0 (x − t) 4 Using the recurrence relation we set 32 7 4 3 u0 (x) = 1 − 2x − x 4 + x 4 , 21 3 7 3 x 1 − 2t − 32 t 4 + 43 t 4 21 (7.80) u1 (x) = − dt 1 (x − t) 4 0 4 3 128 5 16 3 32 7 x4 − x4 + x2 − x2 . = 21 3 63 9 3 32 74 x and ± 43 x 4 appear between u0 (x) and u1 (x). By The noise terms ∓ 21 canceling the noise terms from u0 (x) and verifying that the non-canceled term in u0 (x) justiﬁes the equation (7.79), the exact solution is therefore given by u(x) = 1 − 2x. (7.81)

7.4 The Weakly Singular Volterra Equations

251

Moreover, we can easily obtain the exact solution by using the modiﬁed decomposition method. This can be done by splitting the nonhomogeneous 3 32 74 part f (x) into two parts f1 (x) = 1 − 2x and f2 (x) = − 21 x + 43 x 4 . The use of the modiﬁed method is left as an exercise to the reader. Example 7.15 Solve the weakly singular second kind Volterra integral equation x 1 2 x x 3 u(t)dt. (7.82) u(x) = e − 3(e − 1) + x t 1 0 (e − e ) 3 In this example, the modiﬁed decomposition method will be used. We ﬁrst set 1 f1 (x) = ex , f2 (x) = −3(ex − 1) 3 . (7.83) In view of this, we use the modiﬁed recurrence relation x 1 2et x x 3 dt = 0. u0 (x) = e , u1 (x) = −3(e − 1) + x t 1 0 (e − e ) 3 The exact solution is therefore given by u(x) = ex .

(7.84)

(7.85)

Example 7.16 Solve the weakly singular second-type Volterra integral equation 1 1 1 x u(x) = sin x + (cos x − 1) 6 + u(t)dt. (7.86) 6 0 (cos x − cos t) 56 In this example, we will also use the modiﬁed decomposition method. We ﬁrst set 1 f1 (x) = sin x, f2 (x) = (cos x − 1) 6 . (7.87) In view of this, we use the modiﬁed recurrence relation u0 (x) = sin x, 1 x sin t 1 dt = 0. u1 (x) = (cos x − 1) 6 + 6 0 (cos x − cos t) 56 The exact solution is therefore given by u(x) = sin x.

(7.88)

(7.89)

Example 7.17 Solve the weakly singular second kind Volterra integral equation x 1 3 4 9 10 3 3 3 u(x) = x + x − x − x + u(t)dt. (7.90) 2 2 1 4 20 0 (x − t ) 3 In this example, we will also use the modiﬁed decomposition method. We ﬁrst set 3 4 9 10 (7.91) f1 (x) = x + x3 , f2 (x) = − x 3 − x 3 . 4 20

252

7 Abel’s Integral Equation and Singular Integral Equations

In view of this, we use the modiﬁed recurrence relation x 3 4 9 10 t + t3 u0 (x) = x + x3 , u1 (x) = − x 3 − x 3 + dt = 0. (7.92) 2 2 1 4 20 0 (x − t ) 3 The exact solution is therefore given by (7.93) u(x) = x + x3 . Example 7.18 As a ﬁnal example we consider the weakly singular second kind Volterra integral equation x √ 1 √ u(t)dt. (7.94) u(x) = 2 x − x−t 0 Following the discussion in the previous examples we use the recurrence relation √ u0 (x) = 2 x, x √ t √ dt = −πx, u1 (x) = −2 x − t 0 x πt 4 √ u2 (x) = dt = πx3/2 , (7.95) 3 x−t 0 4 x πt3/2 1 √ u3 (x) = − dt = − π 2 x2 , 3 0 2 x−t .. . Because self-canceling noise terms did not appear in the components u0 (x) and u1 (x), we obtained more components. Consequently, the solution u(x) in a series form is given by √ 4 1 (7.96) u(x) = 2 x − πx + πx3/2 − π 2 x2 + · · · , 3 2 and in a closed form by √ u(x) = 1 − eπx erfc( πx), (7.97) where erfc is the complementary error function normally used in probability topics. The deﬁnitions of the error function and the complementary error function can be found in Appendix D.

Exercises 7.4.1 Solve the following weakly singular Volterra equations, where x ∈ [0, T ]. x 9 4 1 1. u(x) = x − x 3 + 2 u(t)dt 4 0 (x − t) 3 x 1 9 5 x3 + 2. u(x) = x − 1 u(t)dt 10 0 (x − t) 3

7.4 The Weakly Singular Volterra Equations x 2 1 3 3. u(x) = sin x + (cos x − 1) 3 + 1 u(t)dt 2 0 (cos x − cos t) 3 x 2 1 2 4. u(x) = cos x − (sin x) 3 + 1 u(t)dt 3 0 (sin x − sin t) 3 x 2 1 3 5. u(x) = e−x + (e−x − 1) 3 + 1 u(t)dt 2 0 (e−x − e−t ) 3 x 1 3 4 6. u(x) = x − x 3 + 1 u(t)dt 2 4 0 (x − t2 ) 3 x 1 8 7 7. u(x) = x3 − x2 + 1 u(t)dt 2 21 0 (x − t2 ) 4 x 1 2 8. u(x) = x3 + 1 u(t)dt 4 3 0 (x − t4 ) 4 x 1 16 21 x4 + 9. u(x) = x5 − 1 u(t)dt 63 0 (x3 − t3 ) 4 x 1 7 5 x + 10. u(x) = 1 u(t)dt 10 0 (x3 − t3 ) 3 x 6 7 1 11. u(x) = ex − (ex − 1) 7 + 1 u(t)dt 6 0 (ex − et ) 7 x 2 1 3 x 12. u(x) = ex (ex + 1) − (e − 1) 3 (3ex + 7) + 1 u(t)dt 10 0 (ex − et ) 3 x 2 3 1 13. u(x) = sin 2x + (3 cos x + 2)(cos x − 1) 3 + 1 u(t)dt 5 0 (cos x − cos t) 3 x 2 3 1 14. u(x) = sinh x − (cosh x − 1) 3 + 1 u(t)dt 2 0 (cosh x − cosh t) 3 x 1 π 15. u(x) = 1 − + 1 u(t)dt 2 2 0 (x − t2 ) 2 x 1 π2 π 16. u(x) = π − + 1− x2 + 1 u(t)dt 2 2 4 0 (x − t2 ) 2

253

7.4.2 The Successive Approximations Method The successive approximations method, or Picard iteration method provides a scheme that can be used for solving initial value problem or integral equation. The method was used before in Chapter 3. This method solves any problem by ﬁnding successive approximations to the solution by starting with an initial guess, called the zeroth approximation. As will be seen, the zeroth approximation is any selective real-valued function that will be used in a recurrence relation to determine the other approximations.

254

7 Abel’s Integral Equation and Singular Integral Equations

In this section, we will study the weakly singular Volterra integral equation of the second kind x 1 √ u(x) = f (x) + u(t)dt. (7.98) x −t 0 The successive approximations method introduces the recurrence relation x 1 √ un (t)dt, n 0, un+1 (x) = f (x) + (7.99) x −t 0 where the zeroth approximation u0 (x) can be any selective real valued function. We always start with an initial guess for u0 (x), mostly we select 0, 1, x for u0 (x), and by using (7.99), several successive approximations uk , k 1 will be determined as x 1 √ u0 (t)dt, u1 (x) = f (x) + x−t 0 x 1 √ u2 (x) = f (x) + (7.100) u1 (t)dt, x −t 0 x 1 √ u3 (x) = f (x) + u2 (t)dt, x −t 0 and so on. As stated before in Chapter 3, the solution u(x) is given by u(x) = lim un+1 (x). n→∞

(7.101)

The successive approximations method will be explained by studying the following illustrative questions. Example 7.19 Solve the weakly singular integral equation by using the successive approximations method x √ 1 π √ u(x) = x + x − u(t)dt. (7.102) 2 x −t 0 For the zeroth approximation u0 (x), we can select (7.103) u0 (x) = 0. The method of successive approximations admits the use of the iteration formula x √ π 1 √ un (t)dt, n 0. (7.104) un+1 (x) = x + x − 2 x −t 0 Substituting (7.103) into (7.104) we obtain √ π u1 (x) = x + x, 2 √ π 2 3 π 2 x+ x u2 (x) = x+ x − , 2 2 3 (7.105) .. . √ π 2 3 2 3 π π2 2 x + x2 + x2 + x − ··· . x+ x − un+1 (x) = 2 2 3 3 4

7.4 The Weakly Singular Volterra Equations

The solution u(x) is given by u(x) = lim un+1 (x) = n→∞

255

√ x.

(7.106)

Example 7.20 Solve the weakly singular integral equation by using the successive approximations method x 4 3 1 √ u(t)dt. (7.107) u(x) = x − x 2 + 3 x −t 0 For the zeroth approximation u0 (x), we can select (7.108) u0 (x) = 0. The method of successive approximations admits the use of the iteration formula x 4 3 1 2 √ (7.109) un+1 (x) = x − x + un (t)dt, n 0. 3 x−t 0 Proceeding as before we set 4 3 u1 (x) = x − x 2 , 3 4 3 4 3 π 2 2 2 + x − x , u2 (x) = x − x 3 3 2 4 3 π 2 8π 5 4 3 π u3 (x) = x − x 2 + x 2 − x2 + x − x2 , 3 3 2 2 15

(7.110)

.. .

4 3 π 4 3 π 2 8π 5 x − x2 + x 2 − x2 + x − x2 + · · · . 3 3 2 2 15 The solution u(x) is given by un+1 (x) =

u(x) = lim un+1 (x) = x. n→∞

(7.111)

Example 7.21 Solve the weakly singular integral equation x √ 4 3 1 2 √ u(t)dt. u(x) = 1 + x − 2 x − x + 3 x−t 0 For the zeroth approximation u0 (x), we can select u0 (x) = 0. Proceeding as before, we use of the iteration formula x √ 4 3 1 √ un+1 (x) = 1 + x − 2 x − x 2 + un (t)dt, n 0. 3 x−t 0 This in turn gives

(7.112)

(7.113)

256

7 Abel’s Integral Equation and Singular Integral Equations

√ u1 (x) = (1 + x) − 2 x + √ u2 (x) = (1 + x) − 2 x +

4 3 x2 , 3 √ 4 3 4 3 π 2 2 + 2 x+ x − πx + x2 , x 3 3 2

.. .

π π un+1 (x) = (1 + x) − πx + x2 + πx + x2 + · · · . 2 2 The solution u(x) is given by u(x) = lim un+1 (x) = 1 + x. n→∞

(7.114) (7.115)

Example 7.22 We ﬁnally consider the weakly singular Volterra integral equation x √ 1 √ u(t)dt. u(x) = 2 x − x−t 0 We select u0 (x) = 0, hence we use the recurrence relation √ u1 (x) = 2 x, x √ √ u0 (t) √ u2 (x) = 2 x − dt = 2 x − πx, x−t 0 .. . √ 4 3 1 un+1 (x) = 2 x − πx + x 2 − π 2 x2 + · · · . 3 2 This in turn gives √ u(x) = 1 − eπx erfc( πx).

(7.116)

(7.117)

(7.118)

Exercises 7.4.2 Use the successive approximations method to solve the following weakly singular Volterra equations: x √ 1 √ 1. u(x) = x − 2πx + 4 u(t)dt x−t 0 x 5 1 5 √ πx3 + u(t)dt 2. u(x) = x 2 − 16 x−t 0 x √ 1 √ u(t)dt 3. u(x) = 3 − 6 x + x −t 0 x √ 1 1 √ u(t)dt 4. u(x) = 1 − x − πx + 2 x−t 0 x 3 1 3 √ u(t)dt 5. u(x) = x 2 − πx2 + 8 x−t 0 x 1 32 7 √ x2 + u(t)dt 6. u(x) = x3 − 33 x −t 0

7.4 The Weakly Singular Volterra Equations x √ 1 16 5 7. u(x) = 1 + x2 − 2 x − √ x2 + u(t)dt 15 x−t 0 x √ 1 4 3 √ u(t)dt 8. u(x) = 1 − x − 2 x + x 2 + 3 x −t 0 x 1 16 5 √ x2 + u(t)dt 9. u(x) = x2 − 15 x −t 0 x √ 4 1 π √ x + x(1 − x) + u(t)dt 10. u(x) = 1 − 2 3 x−t 0 x √ 1 √ u(t)dt 11. u(x) = π(1 − 2 x) + x −t 0 x 4 3 1 4 u(t)dt 12. u(x) = x(1 + x) − x 2 1 + x + √ 3 5 x−t 0

257

7.4.3 The Laplace Transform Method The Laplace transform method is used before, and the properties were presented in details in Chapters 1, 3, 5, and in the previous section. Recall that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by L{(f1 ∗ f2 )(x)} = F1 (s)F2 (s). (7.119) We will focus our study on the generalized weakly singular Volterra equation of the form x 1 u(t)dt, 0 < α < 1. (7.120) u(x) = f (x) + α 0 (x − t) Taking Laplace transforms of both sides of (7.120) leads to (7.121) L(u(x)) = L(f (x)) + L(x−α )L(u(x)), or equivalently Γ(1 − α) U (s) = F (s) + U (s), (7.122) s1−α that gives s1−α F (s) , (7.123) U (s) = 1−α s − Γ(1 − α) where Γ is the gamma function, U (s) = L{u(x)}, and F (s) = L{f (x)}. In Appendix D, the deﬁnition of the gamma function and some of the relations related to it are given. Applying L−1 to both sides of (7.123) gives the formula s1−α F (s) −1 , (7.124) u(x) = L s1−α − Γ(1 − α) that will be used for the determination of the solution u(x). Notice that the formula (7.124) will be used for solving the weakly singular integral equation, and it is not necessary to use Laplace transform method for each problem.

258

7 Abel’s Integral Equation and Singular Integral Equations

The following weakly singular Volterra equations will be examined by using the Laplace transform method. Example 7.23 Use the Laplace transform method to solve the weakly singular Volterra integral equation x √ 1 √ u(x) = 1 − 2 x + u(t)dt. (7.125) x−t 0 √ Notice that f (x) = 1 − 2 x and α = 12 . This means that √ Γ( 32 ) 1 π 1 (7.126) F (s) = − 2 3 = − 3 , s s s2 s2 √ where we used Γ( 32 ) = 12 Γ( 12 ) = 2π . Substituting (7.126) into the formula given in (7.124) gives √ ⎞ ⎛ 1 s 2 1s − 3π 1 −1 ⎝ −1 s2 ⎠ =L u(x) = L = 1. (7.127) √ 1 s 2 s − π Example 7.24 Use the Laplace transform method to solve the weakly singular Volterra integral equation x 1 4 3 √ u(x) = x − x 2 + u(t)dt. (7.128) 3 x −t 0 3

Notice that f (x) = x − 43 x 2 and α = 12 . This means that √ 1 4 Γ( 52 ) 1 π F (s) = 2 − = 2− 5 , (7.129) s 3 s 52 s 2 s √ where we used Γ( 52 ) = 34 Γ( 12 ) = 34 π. Substituting (7.129) into (7.124) gives ⎛ 1 ⎞ √ s 2 s12 − 5π

s2 ⎠ = L−1 12 = x. (7.130) u(x) = L−1 ⎝ 1 √ s s2 − π

Example 7.25 Use the Laplace transform method to solve the weakly singular Volterra integral equation x 1 9 5 u(x) = x − x 3 + (7.131) 1 u(t)dt. 10 0 (x − t) 3 Notice that f (x) = x −

9 53 x 10

and α = 13 . This means that

Γ( 23 ) 9 Γ( 83 ) 1 1 − = − , (7.132) 8 8 s2 10 s 3 s2 s3 2 where we used Γ( 83 ) = 10 9 Γ( 3 ). Substituting (7.132) into (7.124) gives F (s) =

7.4 The Weakly Singular Volterra Equations

⎛ u(x) = L−1 ⎝

2

s3

1 s2 2 s3

−

Γ( 2 ) 3 8 s3

−Γ( 23 )

259 ⎞

⎠ = L−1

1 s2

= x.

(7.133)

Exercises 7.4.3 In Exercises 1–8, use the Laplace transform method to solve the following weakly singular Volterra equations: x √ 1 √ u(t)dt 1. u(x) = x − 2πx + 4 x −t 0 x 1 16 5 √ x2 + u(t)dt 2. u(x) = x2 − 15 x −t 0 x √ 1 √ u(t)dt 3. u(x) = 4 − 8 x + x−t 0 x 1 32 7 √ x2 + u(t)dt 4. u(x) = x3 − 33 x−t 0 x √ 1 1 √ u(t)dt 5. u(x) = 1 − x − πx + 2 x −t 0 x √ 1 4 3 u(t)dt 6. u(x) = 1 + x − 2 x − x 2 + √ 3 x−t 0 x 3 1 3 u(t)dt 7. u(x) = x 2 − πx2 + √ 8 x−t 0 x √ π 1 4 u(t)dt 8. u(x) = (1 − )x + x(1 − x) + √ 2 3 x −t 0 In Exercises 9–16, use the Laplace transform method to solve the generalized weakly singular Volterra equations: x 9 4 1 9. u(x) = x − x 3 + 2 u(t)dt 4 0 (x − t) 3 x 27 8 1 x3 + 10. u(x) = x2 − 1 u(t)dt 40 0 (x − t) 3 x 1 16 7 11. u(x) = x − x4 + 1 u(t)dt 21 0 (x − t) 4 x 1 243 11 12. u(x) = x3 − x3 + 1 u(t)dt 440 0 (x − t) 3 x 1 9 5 3 2 x3 − x3 + 13. u(x) = 1 + x − 1 u(t)dt 10 2 0 (x − t) 4 x 36 11 1 x6 + 14. u(x) = x − 1 u(t)dt 55 0 (x − t) 6 x 1 25 7 x5 + 15. u(x) = x − 3 u(t)dt 14 0 (x − t) 5 x 1 128 9 16. u(x) = x2 − x4 + 3 u(t)dt 45 0 (x − t) 4

260

7 Abel’s Integral Equation and Singular Integral Equations

References 1. V. Singh, R. Pandey and O. Singh, New stable numerical solutions of singular integral equations of Abel type by using normalized Bernstein polynomials, Appl. Math. Sciences, 3 (2009) 241–255. 2. A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 3. M. Masujima, Applied Mathematical Methods in Theoretical Physics, Wiley, Weinheim, (2005). 4. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 5. R. Estrada and R. Kanwal, Singular Integral Equations, Birkhauser, Berlin, (2000). 6. R. P. Kanwal, Linear Integral Equations, Birkhauser, Boston, (1997). 7. I.N. Sneddon, Mixed boundary value problems in potential theory, Wiley, New York, (1966). 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).

Chapter 8

Volterra-Fredholm Integral Equations

8.1 Introduction The Volterra-Fredholm integral equations [1–2] arise from parabolic boundary value problems, from the mathematical modelling of the spatio-temporal development of an epidemic, and from various physical and biological models. The Volterra-Fredholm integral equations appear in the literature in two forms, namely x b K1 (x, t)u(t)dt + λ2 K2 (x, t)u(t)dt, (8.1) u(x) = f (x) + λ1 0

a

and the mixed form

x

b

u(x) = f (x) + λ

K(r, t)u(t)dtdr, 0

(8.2)

a

where f (x) and K(x, t) are analytic functions. It is interesting to note that (8.1) contains disjoint Volterra and Fredholm integrals, whereas (8.2) contains mixed Volterra and Fredholm integrals. Moreover, the unknown functions u(x) appears inside and outside the integral signs. This is a characteristic feature of a second kind integral equation. If the unknown functions appear only inside the integral signs, the resulting equations are of ﬁrst kind. Examples of the two types of the Volterra-Fredholm integral equations of the second kind are given by x 1 xu(t)dt − tu(t)dt, (8.3) u(x) = 6x + 3x2 + 2 − 0

and u(x) = x +

17 2 x − 2

0

0

x 0

1

(r − t)u(t)drdt.

(8.4)

However, the Volterra-Fredholm integro-diﬀerential equations that appear in scientiﬁc applications will be presented in the next chapter.

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

262

8 Volterra-Fredholm Integral Equations

8.2 The Volterra-Fredholm Integral Equations In this section, we will study some of the reliable methods that will be used for analytic treatment of the Volterra-Fredholm integral equations [1–2] of the form b x K1 (x, t)u(t)dt + K2 (x, t)u(t)dt. (8.5) u(x) = f (x) + 0

a

This type of equations will be handled by using the Taylor series method and the Adomian decomposition method combined with the noise terms phenomenon or the modiﬁed decomposition method. Other approaches exist in the literature but will not be presented in this text.

8.2.1 The Series Solution Method The series solution method was examined before in this text . A real function u(x) is called analytic if it has derivatives of all orders such that the generic form of Taylor series at x = 0 can be written as ∞ a n xn . (8.6) u(x) = n=0

In this section we will apply the series solution method [3–4], that stems mainly from the Taylor series for analytic functions, for solving VolterraFredholm integral equations. We will assume that the solution u(x) of the Volterra-Fredholm integral equation b x K1 (x, t)u(t)dt + K2 (x, t)u(t)dt, (8.7) u(x) = f (x) + 0

a

is analytic, and therefore possesses a Taylor series of the form given in (8.6), where the coeﬃcients an will be determined recurrently. In this method, we usually substitute the Taylor series (8.6) into both sides of (8.7) to obtain ∞ x ∞ k k ak x = T (f (x)) + K1 (x, t) ak t dt k=0

0

b

K2 (x, t)

+

∞

a

2

a0 + a1 x + a2 x + · · · = T (f (x)) + + a

ak t

k

(8.8)

dt,

k=0

or for simplicity we use

k=0

b

x 0

K1 (x, t) a0 + a1 t + a2 t2 + · · · dt

K2 (x, t) a0 + a1 t + a2 t2 + · · · dt,

(8.9)

8.2 The Volterra-Fredholm Integral Equations

263

where T (f (x)) is the Taylor series for f (x). The Volterra-Fredholm integral equation (8.7) will be converted to a regular integral in (8.8) or (8.9) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integrals in (8.8) or (8.9), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (8.6). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. Example 8.1 Solve the Volterra-Fredholm integral equation by using the series solution method x 1 u(x) = −5 − x + 12x2 − x3 − x4 + (x − t)u(t)dt + (x + t)u(t)dt. (8.10) 0

0

Substituting u(x) by the series u(x) =

∞

an xn ,

(8.11)

n=0

into both sides of Eq. (8.10) leads to ∞ an xn = −5 − x + 12x2 − x3 − x4 + n=0

+

1

(x + t)

0

∞

an t

n

x 0

(x − t)

∞

an t

n=0

n

dt (8.12)

dt.

n=0

Evaluating the integrals at the right side, using few terms from both sides, and collecting the coeﬃcients of like powers of x, we ﬁnd a0 + a1 x + a2 x2 + a3 x3 + · · · 1 1 1 1 1 = −5 + a0 + a1 + a2 + a3 + a4 2 3 4 5 6 1 1 1 1 + −1 + a0 + a1 + a2 + a3 + a4 x 2 3 4 5 1 1 + 12 + a0 x2 + −1 + a1 x3 2 6

264

8 Volterra-Fredholm Integral Equations

1 + −1 + a2 x4 + · · · . (8.13) 12 Equating the coeﬃcients of like powers of x in both sides of (8.13), and solving the resulting system of equations, we obtain a0 = 0, a1 = 6, (8.14) a2 = 12, a3 = a4 = a5 = · · · = 0. The exact solution is therefore given by u(x) = 6x + 12x2 .

(8.15)

Example 8.2 Use the series solution method to solve the Volterra-Fredholm integral equation x 1 2 3 5 tu(t)dt + (x + t)u(t)dt. (8.16) u(x) = 2 − x − x − 6x + x + 0

−1

Substituting u(x) by the series u(x) =

∞

an xn ,

(8.17)

n=0

into both sides of Eq. (8.16) leads to ∞ n 2 3 5 an x = 2 − x − x − 6x + x + n=0

1

+

(x + t)

−1

∞

a n tn

x

0

dt.

t

∞

an tn dt

n=0

(8.18)

n=0

Evaluating the integrals at the right side, using few terms from both sides, and collecting the coeﬃcients of like powers of x, we ﬁnd a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · 2 2 = 2 + a1 + a3 3 5 2 2 + −1 + 2a0 + a2 + a4 x 3 5 1 1 2 + −1 + a0 x + −6 + a1 x3 2 3 1 1 + a2 x4 + 1 + a3 x5 + · · · . (8.19) 4 5 Equating the coeﬃcients of like powers of x in both sides of (8.19), and solving the resulting system of equations, we obtain a0 = 2, a1 = 3, a2 = 0, a3 = −5, (8.20) a4 = a5 = · · · = 0. The exact solution is therefore given by

8.2 The Volterra-Fredholm Integral Equations

265

u(x) = 2 + 3x − 5x3 .

(8.21)

Example 8.3 Solve the Volterra-Fredholm integral equation by using the series solution method x 1 x u(x) = e − 1 − x + u(t)dt + xu(t)dt. (8.22) 0

0

7

x

Using the Taylor polynomial for e up to x , substituting u(x) by the Taylor polynomial 7 u(x) = an xn , (8.23) n=0

and proceeding as before leads to 7 7 1 n an x an x = 2a0 + n+1 n=0 n=1 +

1 + a1 2 1 + 2!a2 3 1 + 3!a3 4 x + x + x 2! 3! 4!

1 + 4!a4 5 1 + 5!a5 6 1 + 6!a6 7 x + x + x + · · · . (8.24) 5! 6! 7! Equating the coeﬃcients of like powers of x in both sides of (8.19), and proceeding as before, we obtain 1 a0 = 0, a1 = 1, a2 = 1, a3 = , 2! (8.25) 1 1 1 a4 = , a5 = , a6 = . 3! 4! 5! The exact solution is given by (8.26) u(x) = xex . +

Example 8.4 Solve the Volterra-Fredholm integral equation by using the series solution method π x (x − t)u(t)dt + u(t)dt. (8.27) u(x) = 1 − 0

0

Substituting u(x) by the Taylor polynomial 8 u(x) = an xn ,

(8.28)

n=0

and proceeding as before, we obtain a0 = 1, a2 = −

a1 = a3 = a5 = a7 = 0, 1 , 21

a4 =

1 , 4!

(8.29)

266

8 Volterra-Fredholm Integral Equations

1 1 , a8 = . 6! 8! Consequently, the exact solution is given by u(x) = cos x. a6 = −

(8.30)

Exercises 8.2.1 Use the series solution method to solve the following Volterra-Fredholm integral equations x 1 (x − t)u(t)dt + (x + t)u(t)dt 1. u(x) = −4 − 3x + 11x2 − x3 − x4 + 2

0

3

2. u(x) = −1 − 3x − 2x +

x

(x − t)u(t)dt +

0

3. u(x) = 4 − x − 4x2 − x3 +

x 0

x 0

8. u(x) = −2x + x cos x + 9. u(x) = 2x −

x

0

0

x 0

0

π 2

12. u(x) = −2 − 2x + 2ex +

0

(x + t + 1)u(t)dt

0 1

(x + t)u(t)dt

0

1 0

(4xt2 + 3x2 t)u(t)dt

0

tu(t)dt

π 0

xu(t)dt

x

1

1

(x − t)u(t)dt −

u(t)dt +

(x + t − 1)u(t)dt

xu(t)dt

x 0

1

0

(x − t)u(t)dt +

xu(t)dt +

(x − t)u(t)dt −

11. u(x) = −x ln(1 + x) +

x

tu(t)dt +

(x − t)u(t)dt +

x

10. u(x) = 2 + x − 2 cos x −

(x + t)u(t)dt

x 0

0

0

(3xt + 4x2 t2 )u(t)dt +

7. u(x) = ex − xex + x − 1 +

x

0

5. u(x) = −6 − 2x + 19x3 − x5 + 6. u(x) = −x2 − x4 − x6 +

1

(x − t + 1)u(t)dt +

4. u(x) = −8 − 6x + 11x2 − x4 +

0

π 2

0

xu(t)dt

e−1

0

xu(t)dt

(x − t)u(t)dt +

1

0

xu(t)dt

8.2.2 The Adomian Decomposition Method The Adomian decomposition method [5–8] (ADM) was introduced thoroughly in this text for handling independently Volterra and Fredholm integral equations. The method consists of decomposing the unknown function u(x) of any equation into a sum of an inﬁnite number of components deﬁned by the decomposition series

8.2 The Volterra-Fredholm Integral Equations

u(x) =

∞

267

un (x),

(8.31)

n=0

where the components un (x), n 0 are to be determined in a recursive manner. To establish the recurrence relation, we substitute the decomposition series into the Volterra-Fredholm integral equation (8.5) to obtain ∞ ∞ x b ∞ un (x) = f (x) + K1 (x, t) un (t) dt + K2 (x, t) un (t) dt. 0

n=0

a

n=0

n=0

(8.32) The zeroth component u0 (x) is identiﬁed by all terms that are not included under the integral sign. Consequently, we set the recurrence relation u0 (x) = f (x), x b (8.33) K1 (x, t)un (t)dt + K2 (x, t)un (t)dt, n 0. un+1 (x) = 0

a

Having determined the components u0 (x), u1 (x), u2 (x), . . ., the solution in a series form is readily obtained upon using (8.31). The series solution converges to the exact solution if such a solution exists. We point here that the noise terms phenomenon and the modiﬁed decomposition method will be employed in this section. This will be illustrated by using the following examples. Example 8.5 Use the Adomian decomposition method to solve the following VolterraFredholm integral equation x 1 x u(x) = e + 1 + x + (x − t)u(t)dt − ex−t u(t)dt. (8.34) 0

0

Using the decomposition series (8.31), and using the recurrence relation (8.33) we obtain u0 (x) = ex + 1 + x, x 1 u1 (x) = (x − t)u0 (t)dt + ex−t u0 (t)dt (8.35) 0

0

1 = −x − 1 + x2 + · · · , 2 and so on. We notice the appearance of the noise terms ±1 and ±x between the components u0 (x) and u1 (x). By canceling these noise terms from u0 (x), the non-canceled term of u0 (x) gives the exact solution u(x) = ex , (8.36) that satisﬁes the given equation. It is to be noted that the modiﬁed decomposition method can be applied here. Using the modiﬁed recurrence relation u0 (x) = ex , x 1 (8.37) u1 (x) = 1 + x + (x − t)u0 (t)dt − ex−tu0 (t)dt = 0. 0

0

268

8 Volterra-Fredholm Integral Equations

The exact solution u(x) = ex follows immediately. Example 8.6 Use the modiﬁed Adomian decomposition method to solve the following Volterra-Fredholm integral equation x 1 1 1 1 u(x) = x2 − x4 − − x + (x − t)u(t)dt + (x + t)u(t)dt. (8.38) 12 4 3 0 0 Using the modiﬁed decomposition method gives the recurrence relation 1 u0 (x) = x2 − x4 , 12 1 x 1 1 (8.39) (x − t)u(t)dt + (x + t)u(t)dt u1 (x) = − − x + 4 3 0 0 1 1 4 1 6 1 x − x − x− . = 12 360 60 72 1 4 and so on. We notice the appearance of the noise terms ± 12 x between the components u0 (x) and u1 (x). By canceling the noise term from the u0 (x), the non-canceled term gives the exact solution u(x) = x2 , (8.40) that satisﬁes the given equation. Example 8.7 Use the modiﬁed Adomian decomposition method to solve the following Volterra-Fredholm integral equation π x u(t)dt + (x − t)u(t)dt. (8.41) u(x) = cos x − sin x − 2 + 0

0

Using the modiﬁed decomposition method gives the recurrence relation u0 (x) = cos x, x π (8.42) u(t)dt + (x − t)u(t)dt = 0. u1 (x) = − sin x − 2 + 0

0

Consequently, the exact solution is given by u(x) = cos x.

(8.43)

Example 8.8 Use the modiﬁed Adomian decomposition method to solve the following Volterra-Fredholm integral equation x 1 2 3 4 tu(t)dt + tu(t)dt. (8.44) u(x) = 3x + 4x − x − x − 2 + 0

−1

Using the modiﬁed decomposition method gives the recurrence relation u0 (x) = 3x + 4x2 − x3 , x 4 u1 (x) = −x − 2 + tu(t)dt + 0

1

−1

tu(t)dt

(8.45)

8.3 The Mixed Volterra-Fredholm Integral Equations

269

2 1 = − − x5 + x3 . 5 5 Canceling the noise term −x3 from u0 (x) gives the exact solution u(x) = 3x + 4x2 .

(8.46)

Exercises 8.2.2 Use the modiﬁed decomposition method to solve the following Volterra-Fredholm integral equations x 1 1 1. u(x) = x − x3 + tu(t)dt + t2 u(t)dt 3 0 −1 x π 4 u(t)dt + u(t)dt 2. u(x) = sec2 x − tan x − 1 + 3. u(x) = sin x − cos x − 4. u(x) = x3 −

0

x 0

u(t)dt +

1 5 1 1 x − x− + 20 4 5

x

0

0

π 2

0

u(t)dt

(x − t)u(t)dt +

1

(x + t)u(t)dt

0

x 1 9 5 1 1 (x + t)u(t)dt + (x − t)u(t)dt x − x+ + 20 4 5 0 0 x π 2 6. u(x) = sin x + cos x − x + u(t)dt + (x − t)u(t)dt

5. u(x) = x3 −

7. u(x) = cos x − sin x − 2 +

0 x

u(t)dt +

0

8. u(x) = cos x + (1 − x) sin x − 2 + 9. u(x) = tan x − ln(cos x) + 10. u(x) = cot x − ln(sin x) −

x

0

(x − t)u(t)dt

0

(x − 1)u(t)dt +

x 0

0 π

u(t)dt +

1 x ln(2) + 2

π 4

−π

0

(x − t)u(t)dt

xu(t)dt

4

x π 2

π

u(t)dt +

π 2 π 4

xu(t)dt

x π 1 1 2 1 x − + u(t)dt u(t)dt + π 2 2 0 4 2 x π 1 1 12. u(x) = x + cos x − sin x − x2 + 1 + u(t)dt u(t)dt + π 2 0 4 11. u(x) = x + sin x + cos x −

2

8.3 The Mixed Volterra-Fredholm Integral Equations In a parallel manner to our analysis in the previous section, we will study the mixed Volterra-Fredholm integral equations [2], given by

270

8 Volterra-Fredholm Integral Equations

x

b

u(x) = f (x) +

K(r, t)u(t)dtdr, 0

(8.47)

a

where f (x) and K(x, t) are analytic functions. It is interesting to note that (8.47) contains mixed Volterra and Fredholm integral equations. The Fredholm integral is the interior integral, whereas the Volterra integral is the exterior one. Moreover, the unknown function u(x) appears inside and outside the integral signs. This is a characteristic feature of a second kind integral equation. This type of equations will be handled by using the series solution method and the Adomian decomposition method, including the use of the noise terms phenomenon or the modiﬁed decomposition method. Other approaches exist in the literature but will not be presented in this text.

8.3.1 The Series Solution Method The series solution method was examined before in this chapter. A real function u(x) is called analytic if it has derivatives of all orders such that the generic form of Taylor series at x = 0 can be written as ∞ an xn . (8.48) u(x) = n=0

We will assume that the solution u(x) of the mixed Volterra-Fredholm integral equation x b K(r, t)u(t)dtdr, (8.49) u(x) = f (x) + 0

a

is analytic, and therefore possesses a Taylor series of the form given in (8.48), where the coeﬃcients an will be determined in an algebraic manner. In this method, we usually substitute the Taylor series (8.48) into both sides of (8.49) to obtain ∞ x b ∞ k k ak x = T (f (x)) + K(r, t) ak t dtdr, (8.50) 0

k=0

or for simplicity we use 2

a0 + a1 x + a2 x + · · · = T (f (x)) +

a

k=0

x b 0

a

K(r, t) a0 + a1 t + a2 t2 + · · · dtdr,

(8.51) where T (f (x)) is the Taylor series for f (x). The mixed Volterra-Fredholm integral equation (8.49) will be converted to a traditional integral in (8.50) or (8.51) where instead of integrating the unknown function u(x), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used.

8.3 The Mixed Volterra-Fredholm Integral Equations

271

We ﬁrst integrate the inner integral, and then we integrate the outer integral in (8.50) or (8.51), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine a system of algebraic equations in aj , j 0. Solving this system of equations will lead to a complete determination of the coefﬁcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (8.48). The exact solution may be obtained if such an exact solution exists, otherwise a truncated series is used for numerical approximations. Example 8.9 Solve the mixed Volterra-Fredholm integral equation by using the Taylor series solution method x 1 17 (r − t)u(t)dtdr. (8.52) u(x) = 11x + x2 + 2 0 0 Substituting u(x) by the series ∞ u(x) = an xn , (8.53) n=0

into both sides of Eq. (8.52) leads to x 1 ∞ ∞ 17 2 n n (r − t) dtdr. an x = 11x + x + an t 2 0 0 n=0 n=0

(8.54)

Evaluating the integrals at the right side, using few terms from both sides, collecting the coeﬃcients of like powers of x, and equating the coeﬃcients of like powers of x in both sides, we obtain a0 = 0, a1 = 6, (8.55) a2 = 12, a3 = a4 = a5 = · · · = 0. The exact solution is therefore given by u(x) = 6x + 12x2 .

(8.56)

Example 8.10 Solve the mixed Volterra-Fredholm integral equation by using the series solution method x 1 9 (r − t)u(t)dtdr. (8.57) u(x) = 2 + 4x − x2 − 5x3 + 8 0 0 Substituting u(x) by the series ∞ u(x) = an xn , (8.58) n=0

into both sides of Eq. (8.57) leads to x 1 ∞ ∞ 9 2 n 3 n (r − t) dt dr. (8.59) an x = 2 + 4x − x − 5x + an t 8 0 0 n=0 n=0

272

8 Volterra-Fredholm Integral Equations

Evaluating the integrals at the right side, using few terms from both sides, collecting the coeﬃcients of like powers of x, and equating the coeﬃcients of like powers of x in both sides, we obtain a0 = 2, a1 = 3, a2 = 0, a3 = −5, (8.60) a4 = a5 = · · · = 0. The exact solution is therefore given by (8.61) u(x) = 2 + 3x − 5x3 . Example 8.11 Solve the mixed Volterra-Fredholm integral equation by using the series solution method x 1 1 ru(t)dtdr. (8.62) u(x) = xex − x2 + 2 0 0 Using the Taylor polynomial for xex up to x9 , substituting u(x) by the Taylor polynomial 9 an xn , (8.63) u(x) = n=0

and proceeding as before, we obtain a0 = 0, a1 = 1, a2 = 1, 1 1 a4 = , a3 = , 2! 3! .. . The exact solution is therefore given by u(x) = xex .

a5 =

1 , 4!

(8.64)

(8.65)

Example 8.12 Solve the mixed Volterra-Fredholm integral equation by using the series solution method x π2 π u(x) = cos x + sin x − x2 + x + (r − t)u(t)dt dr. (8.66) 2 0 0 Proceeding as before we ﬁnd 1 a1 = 1, a2 = − , a0 = 1, 2! 1 1 1 (8.67) a 3 = − , a4 = , a5 = , 3! 4! 5! 1 1 1 a6 = − , a7 = − , a8 = 6! 7! 8! and so on. The series solution is given by 1 1 1 1 (8.68) u(x) = 1 − x2 + x4 + · · · + x − + x5 + · · · , 2! 4! 3! 5!

8.3 The Mixed Volterra-Fredholm Integral Equations

273

that converge to the exact solution u(x) = cos x + sin x.

(8.69)

Exercises 8.3.1 Use the series solution method to solve the following mixed Volterra-Fredholm integral equations x 1 15 2 x + 1. u(x) = 2 + 12x + (r − t)u(t)dtdr 2 0 0 x 1 (r − t)u(t)dtdr 2. u(x) = 6 + 19x − 6x2 + 3. u(x) = 2 + 10x − 2x2 + 4. u(x) = 2 + 6x2 + 5. u(x) = 7x +

x

0

31 2 x + 3

0

−1 x

6. u(x) = 4 + 14x − 2x2 +

0 1

0

0

x

−1

(r − t)u(t)dtdr

(r − t)u(t)dtdr

1 0

1

x

(r 2 − t2 )u(t)dtdr

0

1

−1

7. u(x) = 6 + 9x + 2x2 − 2x3 +

(r 3 − t3 )u(t)dtdr

x 0

1

−1

(rt2 + r2 t)u(t)dtdr

x π 1 2 2 x + (r − t)u(t)dtdr 2 0 0 x e−1 u(t)dtdr 9. u(x) = ln(1 + x) − x + 8. u(x) = sin x + x −

0

0

x e−1 1 10. u(x) = ln(1 + x) − x2 + ru(t)dtdr 2 0 0 x π 2 11. u(x) = cos x + 2 u(t)dtdr π 0 0 x π 1 u(t)dtdr 12. u(x) = tan x − x + ln 2 0 0

8.3.2 The Adomian Decomposition Method The Adomian decomposition method (ADM) was introduced thoroughly in this text for handling independently Volterra integral equations, Fredholm integral equations, and the Volterra-Fredholm integral equations. The method consists of decomposing the unknown function u(x) of any equation into a sum of an inﬁnite number of components deﬁned by the decomposition series

274

8 Volterra-Fredholm Integral Equations

u(x) =

∞

un (x).

(8.70)

n=0

To establish the recurrence relation, we substitute decomposition series into the mixed Volterra-Fredholm integral equation (8.49) to obtain ∞ x b ∞ un (x) = f (x) + K(r, t) un (t) dt dr. (8.71) 0

n=0

a

n=0

Consequently, we set the recurrence relation u0 (x) = f (x), x b un+1 (x) = K(r, t)un (t)dtdr, 0

n 0.

(8.72)

a

The solution in a series form is readily obtained upon using (8.70). The series may converge to the exact solution if such a solution exists. We point out here that the noise terms phenomenon and the modiﬁed decomposition method will be employed heavily. Example 8.13 Solve the mixed Volterra-Fredholm integral equation by using the Adomian decomposition method x 1 1 2 x ru(t)dtdr. (8.73) u(x) = xe − x + 2 0 0 Using the Adomian decomposition method we set the recurrence relation 1 u0 (x) = xex − x2 , 2 x 1 (8.74) ruk (t)dtdr, k 0. uk+1 (x) = 0

0

This in turn gives 1 u0 (x) = xex − x2 , 2 x 1 5 2 u1 (x) = x , ru0 (t)dtdr = 12 0 0 x 1 5 2 ru1 (t)dtdr = x , u2 (x) = 72 0 0 x 1 5 2 ru2 (t)dtdr = u3 (x) = x , 432 0 0 and so on. Using (8.70) gives the series solution 1 2 5 2 1 1 x u(x) = xe − x + x 1 + + + ··· , 2 12 6 36 that converges to the exact solution u(x) = xex ,

(8.75)

(8.76)

(8.77)

8.3 The Mixed Volterra-Fredholm Integral Equations

275

obtained upon ﬁnding the sum of the inﬁnite geometric series. However, we can obtain the exact solution by using the modiﬁed decomposition method. This can be achieved by setting u0 (x) = xex , x 1 (8.78) 1 u1 (x) = − x2 + ru0 (t)dtdr = 0. 2 0 0 Accordingly the other components uj (x), j 2 = 0. This gives the exact solution by u(x) = xex . (8.79) Example 8.14 Solve the mixed Volterra-Fredholm integral equation by using the Adomian decomposition method x π2 1 2 ru(t)dtdr. (8.80) u(x) = sin x − x + 2 0 0 Using the Adomian decomposition method we set the recurrence relation 1 u0 (x) = sin x − x2 , 2 x π2 (8.81) uk+1 (x) = ruk (t)dtdr, k 0. 0

0

This in turn gives 1 u0 (x) = sin x − x2 , 2 x π2 (8.82) 1 π3 2 u1 (x) = x . ru0 (t)dtdr = sin x + x2 − 2 96 0 0 1 2 By canceling the noise term − 2 x from u0 , the exact solution is given by u(x) = sin x. (8.83) This result satisﬁes the given equation. However, we can obtain the exact solution by using the modiﬁed decomposition method. This can be achieved by setting u0 (x) = sin x, x π2 (8.84) 1 ru0 (t)dtdr = 0. u1 (x) = − x2 + 2 0 0 Accordingly the other components uj (x), j 2 = 0. This gives the same solution obtained above. Example 8.15 Solve the mixed Volterra-Fredholm integral equation by using the Adomian decomposition method x π u(x) = cos x − 2x + (r − t)u(t)dtdr. (8.85) 0

0

276

8 Volterra-Fredholm Integral Equations

Using the modiﬁed decomposition method we set u0 (x) = cos x, x π (r − t)u0 (t)dtdr = 0. u1 (x) = −2x + 0

(8.86)

0

Accordingly the other components uj (x), j 2 = 0. This gives the exact solution by u(x) = cos x. (8.87) Example 8.16 Solve the mixed Volterra-Fredholm integral equation by using the Adomian decomposition method x e−1 1 ru(t)dtdr. (8.88) u(x) = ln(1 + x) − x2 + 2 0 0 Using the modiﬁed decomposition method we set u0 (x) = ln(1 + x), x e−1 (8.89) 1 u1 (x) = − x2 + ru0 (t)dtdr = 0. 2 0 0 Accordingly the other components uj (x), j 2 = 0. This gives the exact solution by u(x) = ln(1 + x). (8.90)

Exercises 8.3.2 In Exercises 1–6, use the Adomian decomposition method to solve the mixed VolterraFredholm integral equations x 1 7 1. u(x) = x + (r 3 − t3 )u(t)dtdr 5 0 −1 x 1 2 2. u(x) = x + x3 + (rt2 − r2 t)u(t)dtdr 9 0 −1 x 1 4 3. u(x) = x2 + (rt2 − r2 t)u(t)dtdr 5 0 −1 x 1 2 3 4. u(x) = (rt2 + r2 t)u(t)dtdr x + 15 0 −1 x 1 1 5. u(x) = x3 − x2 + (1 + rt)u(t)dtdr 5 0 −1 x 1 2 6. u(x) = x2 − x + (1 + rt)u(t)dtdr 3 0 −1 In Exercises 7–12, use the modiﬁed decomposition method to solve the following Volterra-Fredholm integral equations x π 1 2 (r − t)u(t)dtdr 7. u(x) = x sin x − x2 + (π − 2)x + 2 0 0

8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables π2 2 x + 4 9. u(x) = ln(1 + x) − x +

8. u(x) = x cos x −

x

0

π

ru(t)dtdr

0

x

0

10. u(x) = tan x − x2 ln 2 +

e−1

u(t)dtdr

0 x

0

277

π 3

0

ru(t)dtdr

x π π 1 4 (r + t)u(t)dtdr − ln 2)x − x2 + 4 2 0 0 x 2 ru(t)dtdr 12. u(x) = 2xex − (e2 + 1)x2 +

11. u(x) = sec2 x − (

0

0

8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables In this section we present a reliable strategy for solving the linear mixed Volterra-Fredholm integral equation of the form t F (x, t, r, s)u(r, s)drds, (x, t) ∈ Ω × [0, T ], (8.91) u(x, t) = f (x, t) + 0

Ω

where u(x, t) is an unknown function, and f (x, t) and F (x, t, r, s) are analytic functions on D = Ω × [0, T ], where Ω is a closed subset of Rn , n = 1, 2, 3. The mixed Volterra-Fredholm integral equation (8.91) in two variables arises from parabolic boundary value problems, the mathematical modeling of the spatio-temporal development of an epidemic, and in various physical and biological models. In the literature, some methods, such as the projection method, time collocation method, the trapezoidal Nystrom method, and Adomian method were used to handle this problem. It was found that these techniques encountered diﬃculties in terms of computational work used, and therefore, approximate solutions were obtained for numerical purposes. In particular, it was found that a complicated term f (x, t) can cause diﬃcult integrations and proliferation of terms in Adomian recursive scheme. To overcome the tedious work of the existing strategies, the modiﬁed decomposition method will form a useful basis for studying the mixed VolterraFredholm integral equation (8.91). The size of the computational work can be dramatically reduced by using the modiﬁed decomposition method. Moreover, the convergence can be accelerated by combining the modiﬁed method with the noise terms phenomenon. The noise terms were deﬁned as the identical terms with opposite signs that may appear in the ﬁrst two components of the series solution of u(x). The modiﬁed decomposition method and the noise terms phenomenon were presented in details in Chapters 3, 4, and 5.

278

8 Volterra-Fredholm Integral Equations

8.4.1 The Modiﬁed Decomposition Method In what follows we give a brief review of the modiﬁed decomposition method. For many cases, the function f (x, t) can be set as the sum of two partial functions, namely f1 (x, t) and f2 (x, t). In other words, we can set (8.92) f (x, t) = f1 (x, t) + f2 (x, t). To minimize the size of calculations, we identify the zeroth component u0 (x, t) by one part of f (x, t), namely f1 (x, t) or f2 (x, t). The other part of f (x, t) can be added to the component u1 (x, t) among other terms. In other words, the modiﬁed decomposition method introduces the modiﬁed recurrence relation u0 (x, t) = f1 (x, t), t u1 (x, t) = f2 (x, t) + F (x, t, r, s)u0 (r, s)drds, (8.93) 0 Ω t F (x, t, r, s)uk (r, s)drds, k 1. uk+1 (x, t) = Ω

0

If noise terms appear between u0 (x, t) and u1 (x, t), then by canceling these terms from u0 (x, t), the remaining non-canceled terms of u0 (x, t) may give the exact solution. This can be satisﬁed by direct substitution. In what follows, we study some illustrative examples. Example 8.17 Use the modiﬁed decomposition method to solve the mixed Volterra-Fredholm integral equation t u(x, t) = f (x, t) + F (x, t, r, s)u(r, s)drds, (x, t) ∈ Ω × [0, T ], (8.94) Ω

0

where

1 f (x, t) = e cos(x) + t cos(x) + t cos(x − 1) sin 1 , 2 F (x, t, r, s) = − cos(x − r)es−t , −t

with Ω = [0, 1]. Substituting the decomposition series ∞ u(x, t) = un (x, t),

(8.95)

(8.96)

n=0

into (8.94) gives ∞ 1 −t un (x, t) = e cos(x) + t cos(x) + t cos(x − 1) sin 1 2 n=0 ∞ t s−t − cos(x − r)e un (r, s) drds. 0

Ω

n=0

As stated before we decompose f (x, t) into two parts as follows:

(8.97)

8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables

279

f0 (x, t) = e−t [cos(x) + t cos(x)], (8.98) 1 −t te cos(x − 1) sin 1. 2 The modiﬁed decomposition technique admits the use of the recursive relation u0 (x, t) = e−t [cos(x) + t cos(x)], 1 u1 (x, t) = te−t cos(x − 1) sin 1 2 t (8.99) − cos(x − r)es−t u0 (r, s)drds f1 (x, t) =

0

t uk+1 (x, t) = −

0

Ω

Ω

cos(x − r)es−t uk (r, s)drds,

k 1.

This gives u0 (x, t) = e−t [cos(x) + t cos(x)], (8.100) 1 1 sin(x + 4) + sin2 2 sin x + · · · u1 (x, t) = −te−t cos(x) + te−t 2 2 −t The self-canceling noise terms te cos(x) and −te−t cos(x) appear between the components u0 (x, t) and u1 (x, t) respectively. By canceling this term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the equation (8.94), the exact solution is given by u(x, t) = e−t cos(x).

(8.101)

Example 8.18 Use the modiﬁed decomposition method to solve the mixed Volterra-Fredholm integral equation t F (x, t, r, s)u(r, s) dr ds, (x, t) ∈ Ω × [0, T ], (8.102) u(x, t) = f (x, t) + 0

where

Ω

π cos x(t cos t − sin t), 4 F (x, t, r, s) = cos(x − r) sin(t − s),

f (x, t) = cos x sin t +

with Ω = [0, π]. Substituting the decomposition series ∞ un (x, t), u(x, t) =

(8.103)

(8.104)

n=0

into (8.102) gives ∞ π un (x, t) = cos x sin t + cos x(t cos t − sin t) 4 n=0 ∞ t cos(x − r) sin(t − s) un (r, s) drds. + 0

Ω

n=0

(8.105)

280

8 Volterra-Fredholm Integral Equations

As stated before we decompose f (x, t) into two parts as follows: π f0 (x, t) = cos x sin t + t cos x cos t, 4 (8.106) π f1 (x, t) = − cos x sin t. 4 The modiﬁed decomposition technique admits the use of the recursive relation π u0 (x, t) = cos x sin t + t cos x cos t, 4 t π u1 (x, t) = − cos x sin t + cos(x − r) sin (t − s)u0 (r, s)drds (8.107) 4 0 Ω t cos(x − r) sin (t − s)uk (r, s)drds, k 1. uk+1 (x, t) = 0

Ω

This gives u0 (x, t) = cos x sin t +

π t cos x cos t, 4

(8.108) π u1 (x, t) = − t cos x cos t + · · · . 4 π The self-canceling noise terms 4 t cos x cos t and − π4 t cos x cos t appear between the components u0 (x, t) and u1 (x, t) respectively. By canceling this term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the equation (8.102), the exact solution u(x, t) = cos x sin t, (8.109) follows immediately. Example 8.19 Use the modiﬁed decomposition method to solve the mixed Volterra-Fredholm integral equation t F (x, t, r, s)u(r, s) dr ds, (x, t) ∈ Ω × [0, T ], (8.110) u(x, t) = f (x, t) + 0

Ω

where f (x, t) = sin(x + t) − 4 cos t − π sin t − 2t sin t + 4, F (x, t, r, s) = r + s, with Ω = [0, π]. Substituting the decomposition series ∞ u(x, t) = un (x, t),

(8.111)

(8.112)

n=0

into (8.110) gives ∞ un (x, t) = sin(x + t) − 4 cos t − π sin t − 2t sin t + 4 n=0 ∞ t + (r + s) un (r, s) drds. 0

Ω

n=0

(8.113)

8.4 The Mixed Volterra-Fredholm Integral Equations in Two Variables

281

As stated before we decompose f (x, t) into two parts as follows: f0 (x, t) = sin(x + t) − 4 cos t − π sin t, (8.114) f1 (x, t) = −2t sin t + 4. The modiﬁed decomposition technique admits the use of the recursive relation u0 (x, t) = sin(x + t) − 4 cos t − π sin t, t (r + s) (u0 (r, s)) drds u1 (x, t) = −2t sin t + 4 + (8.115) Ω 0 t uk+1 (x, t) = (r + s)uk (r, s)drds, k 1. 0

This gives

Ω

u0 (x, t) = sin(x + t) − 4 cos t − π sin t, u1 (x, t) = 4 cos t + π sin t + · · · .

(8.116)

By canceling the two noise terms from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the equation (8.110), the exact solution u(x, t) = sin(x + t), (8.117) is readily obtained. Example 8.20 Use the modiﬁed decomposition method to solve the mixed Volterra-Fredholm integral equation t F (x, t, r, s)u(r, s)drds, (x, t) ∈ Ω × [0, T ], (8.118) u(x, t) = f (x, t) + 0

where

Ω

3 + 4x 1 + 2x 3 1 4 t + t2 − t + t , 12 6 6 F (x, t, r, s) = x + t + r − 2s,

f (x, t) = x2 −

(8.119)

with Ω = [0, 1]. Proceeding as before we set ∞ 3 + 4x 1 + 2x 3 1 4 t + t2 − t + t un (x, t) = x2 − 12 6 6 n=0

t + 0

Ω

(x + t + r − 2s)

∞

un (r, s) drds. (8.120)

n=0

Decomposing f (x, t) into two parts as follows: 3 + 4x 1 + 2x 3 1 4 t + t2 , f1 (x, t) = − t + t , f0 (x, t) = x2 − 12 6 6 and using recursive relation

(8.121)

282

8 Volterra-Fredholm Integral Equations

3 + 4x t + t2 , u0 (x, t) = x2 − 12 t u1 (x, t) = v + (x + t + r − 2s)u0 (r, s)drds t uk+1 (x, t) =

0

Ω

0

Ω

(x + t + r − 2s)uk (r, s)drds,

(8.122)

k 1.

This gives 3 + 4x t + t2 , 12 (8.123) 3 + 4x u1 (x, t) = t+ ··· . 12 By canceling the noise term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the equation (8.118), the exact solution u0 (x, t) = x2 −

u(x, t) = x2 + t2 ,

(8.124)

is readily obtained.

Exercises 8.4.1 Use the modiﬁed decomposition method to solve the mixed Volterra-Fredholm integral equations in two variables t F (x, t, r, s)u(r, s)drds, (x, t) ∈ Ω × [0, T ], (8.125) u(x, t) = f (x, t) + 0

Ω

where f (x, t), F (x, t, r, s), and Ω are given by: 1. f = xt −

1 3 1 3 xt + t , F = (x − r)(t − s), Ω = [0, 1] 12 18

2. f = x + t − 3t2 − 4t3 , F = 12s, Ω = [0, 1] 2 π π π2 − t2 + 2t sin t − , F = cos r sin s, Ω = 0, 3. f = x2 − t2 + cos t 4 4 2 2 π π 4. f = x2 − t2 − sin t − t2 + 2t cos t, F = cos r cos s, Ω = 0, 4 2 1 1 t5 x, F = (x − r)(t − s), Ω = [−1, 1] 5. f = 1 + x3 + t2 + t3 − t2 + 5 10 4 3 t + t2 , F = r − s, Ω = [−1, 1] 9 π 7. f = e−t cos x + te−t cos x, F = − cos(x − r)es−t , Ω = [0, π] 2

6. f = x2 t − xt2 +

8. f = cos(x − t) + 4 sin t + cos t(π − 2t) − π, F = r − s, Ω = [0, π] 9. f = xet −

5 t 1 5 e + tet + , F = r − s, Ω = [0, 1] 6 2 6

10. f = cos(x + t) + 2π sin t − 2πt cos t, F = rs, Ω = [−π, π] 11. f = cos x cos t + 2 cos t + 2t sin t − 2, F = rs, Ω = [0, π]

References 12. f = e−t sin x +

283 π −t te sin x, F = − cos(x − r)es−t , Ω = [0, π] 2

References 1. K. Maleknejad and M. Hadizadeh, A New computational method for VolterraFredholm integral equations, Comput. Math. Appl., 37 (1999) 1–8. 2. A.M. Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. Comput., 127 (2002) 405–414. 3. H.T. Davis, Introduction to Nonlinear Diﬀerential and Integral Equations, Dover, New York, (1962). 4. R. Kress, Linear Integral Equations, Springer, Berlin, (1999). 5. G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994). 6. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, San Diego, (1986). 7. A.M.Wazwaz, A reliable modiﬁcation of Adomian decomposition method, Appl. Math. Comput., 92 (1998) 1–7. 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).

Chapter 9

Volterra-Fredholm Integro-Diﬀerential Equations

9.1 Introduction The Volterra-Fredholm integro-diﬀerential equations [1–4] appear in two types, namely: x b (k) K1 (x, t)u(t)dt + λ2 K2 (x, t)u(t)dt, (9.1) u (x) = f (x) + λ1 a

and the mixed form u

(k)

a

x b

(x) = f (x) + λ

K(r, t)u(t)dtdr, 0

(9.2)

a

k

u(x) where u(k) (x) = d dx k . The ﬁrst type contains disjoint integrals and the second type contains mixed integrals such that the Fredholm integral is the interior one, and Volterra is the exterior integral.

9.2 The Volterra-Fredholm Integro-Diﬀerential Equation In this section we will ﬁrst study the Volterra-Fredholm integro-diﬀerential equation (9.1). The Adomian decomposition method can be used after integrating both sides of any equation k times and using the initial conditions. This type of equations will be handled by using the Taylor series solution method and the variational iteration method only.

9.2.1 The Series Solution Method The series solution method [2,5] was examined before in this chapter and in Chapters 3, 4, and 5. The generic form of Taylor series at x = 0 can be A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

286

9 Volterra-Fredholm Integro-Diﬀerential Equations

written as u(x) =

∞

a n xn .

(9.3)

n=0

We will assume that the solution u(x) of the Volterra-Fredholm integrodiﬀerential equation x b K1 (x, t)u(t)dt + K2 (x, t)u(t)dt, (9.4) u(k) (x) = f (x) + 0

a

is analytic, and therefore possesses a Taylor series of the form given in (9.3), where the coeﬃcients an will be determined algebraically. In this method, we usually substitute the Taylor series (9.3) into both sides of (9.4) to obtain ∞ ∞ (k) x n n an x = T (f (x)) + K1 (x, t) an t dt n=0

0

b

K2 (x, t)

+ a

∞

n=0

an t

n

(9.5)

dt,

k=0

or for simplicity we use (a0 + a1 x + a2 x2 + · · · )(k) = T (f (x)) x

K1 (x, t) a0 + a1 t + a2 t2 + · · · dt + 0

+

b

(9.6)

K2 (x, t) a0 + a1 t + a2 t2 + · · · dt,

a

where T (f (x)) is the Taylor series for f (x). The integro-diﬀerential equation (9.4) will be converted to a traditional integral, where terms of the form tn , n 0 will be integrated. We ﬁrst integrate the right side of the integrals in (9.5) or (9.6), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine the coeﬃcients aj , j 0. Solving the resulting equations will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (9.3). The exact solution may be obtained if such an exact solution exists. It is to be noted that the series method works eﬀectively if the solution u(x) is a polynomial. However, if u(x) is any elementary function, more terms of the series are needed to obtain an approximation of high degree of accuracy. This analysis will be tested by discussing the following examples. Example 9.1 Solve the Volterra-Fredholm integro-diﬀerential equation by using the series solution method

9.2 The Volterra-Fredholm Integro-Diﬀerential Equation

u (x) = 11 + 17x − 2x3 − 3x4 +

x

1

tu(t)dt + 0

0

287

(x − t)u(t)dt, u(0) = 0. (9.7)

Substituting u(x) by the series u(x) =

∞

an xn ,

(9.8)

n=0

into both sides of the equation (9.7), and using the initial condition a0 = 0, we ﬁnd ∞ x ∞ n 3 4 n nan x = 11 + 17x − 2x − 3x + an t t dt (9.9) 0

n=1

1

+ 0

(x − t)

∞

n=0

an t

n

dt.

n=0

Evaluating the integrals at the right side, using few terms from both sides, and collecting the coeﬃcients of like powers of x, and equating the coeﬃcients of like powers of x in both sides of the resulting equation, we obtain a0 = 0, a1 = 6, a2 = 12, aj = 0, j 3. (9.10) The exact solution is therefore given by u(x) = 6x + 12x2 .

(9.11)

Example 9.2 Solve the Volterra-Fredholm integro-diﬀerential equation by using the series solution method x 1 2 3 5 u(t)dt + (1 + xt)u(t)dt, u(0) = 1. u (x) = 2 − 5x − 3x − 20x − x + 0

−1

(9.12)

Substituting u(x) by the series u(x) =

∞

an xn ,

(9.13)

n=0

into both sides of Eq. (9.12), and using the initial condition a0 = 1, we ﬁnd ∞ x ∞ n n 2 3 5 nan x = 2 − 5x − 3x − 20x − x + an t dt n=1

0

1

+

(1 + xt) −1

∞

n=0

(9.14)

n

an t dt.

n=0

Evaluating the integrals at the right side, using few terms from both sides, and collecting the coeﬃcients of like powers of x, and equating the coeﬃcients of like powers of x in both sides of the resulting equation, we obtain a0 = 1, a1 = 6, a2 = a3 = 0, a4 = 5, aj = 0, j 5. (9.15) The exact solution is therefore given by

288

9 Volterra-Fredholm Integro-Diﬀerential Equations

u(x) = 1 + 6x + 5x4 .

(9.16)

Example 9.3 Solve the Volterra-Fredholm integro-diﬀerential equation by using the series solution method x 1 u (x) = 2ex − 2 + u(t)dt + u(t)dt, u(0) = 0. (9.17) 0

0

Substituting u(x) by the Taylor polynomial 10 u(x) = an xn ,

(9.18)

n=0

and proceeding as before, we obtain a0 = 0,

a1 = 1,

a2 = 1,

a3 =

1 , 2!

a4 =

1 , 3!

a5 =

1 . 4!

(9.19)

The exact solution is given by u(x) = xex .

(9.20)

Example 9.4 Solve the Volterra-Fredholm integro-diﬀerential equation by using the series solution method x π2 u (x) = 2 sin x − x − 3 (x − t)u(t)dt + u(t)dt, (9.21) 0 0 u(0) = u (0) = 1,

u (0) = −1.

Using the series assumption for u(x) and proceeding as before, we obtain 1 a0 = 1, a1 = 1, a2 = − , 2! (9.22) 1 1 1 a 3 = − , a 4 = , a5 = , 3! 4! 5! and so on. the exact solution is given by u(x) = sin x + cos x.

(9.23)

Exercises 9.2.1 Solve the Volterra-Fredholm integro-diﬀerential equations by using the series solution method x 1 1. u (x) = 6 + 4x − x3 + (x − t)u(t)dt + (1 − xt)u(t)dt, u(0) = 0 0

−1

1 11 2 (t − 2)u(t)dt + (1 − xt)u(t)dt, u(0) = 1 x − 2x3 + 2 0 −1 x 1 3. u (x) = −4 + 6x − x2 − x4 + xu(t)dt + (1 − xt)u(t)dt, u(0) = 1

2. u (x) = 4 + 6x +

0

x

−1

9.2 The Volterra-Fredholm Integro-Diﬀerential Equation 4. u (x) = −4 + 6x + u(0) = 1 5. u (x) = −8 −

1 2 3 x + x3 + x4 + 2 4

u(0) = 1, u (0) = −3 6. u (x) = −500 +

1 4 3 6 x + x + 6 20

250 3 x + 3

x

0

x

0

u(t)dt +

0

tu(t)dt +

0

x

u(t)dt +

0

1

0

π 2

−1

1 −1

(1 − xt)u(t)dt,

(x − t)u(t)dt,

(xt2 − x2 t)u(t)dt,

tu(t)dt, u(0) = 1, u (0) = 1

9. u (x) = −6 − 2x − 3x2 − 4x3 +

x

0

u(t)dt +

u(0) = 2, u (0) = 6, u (0) = −24 x tu(t)dt + 10. u (x) = x − 2x3 + 3x4 + 0

0

0

x

0

π

u(t)dt +

tu(t)dt,

0

xu(t)dt,

u(t)dt, u(0) = 0, u (0) = 1, u (0) = 0 π 2

0

1

1

0

u(0) = 0, u (0) = 6, u (0) = −24 x 11. u (x) = 3 − 4 cos x + u(t)dt + 12. u (x) = −1 − 4 sin x +

1

u(t)dt, u(0) = −1, u (0) = 1

0

0

1

x

u(0) = 69, u (0) = −168 x u(t)dt + 7. u (x) = 2 cos x − 1 + 8. u (x) =

(xt2 − x2 t)u(t)dt +

289

u(t)dt, u(0) = 0, u (0) = 0, u (0) = 2

9.2.2 The Variational Iteration Method The variational iteration method was used before to handle Volterra and Fredholm integral equations. It was discussed before, that the method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists. The variational iteration method will also be used in this section to study the Volterra-Fredholm integro-diﬀerential equations. The standard ith order Volterra-Fredholm integro-diﬀerential equation is of the form x b u(i) (x) = f (x) + K1 (x, t)u(t)dt + K2 (x, t)u(t)dt, (9.24) 0

(i)

di u dxi ,

a

(i−1)

and u(0), u (0), . . . , u (0) are the initial conditions. where u (x) = The right side contains two disjoint integrals. Concerning the kernel K2 (x, t), we will discuss K2 (x, t) being separable and given by K2 (x, t) = g(x)h(t),

(9.25)

290

9 Volterra-Fredholm Integro-Diﬀerential Equations

or may be diﬀerence kernel and given by (9.26) K2 (x, t) = K2 (x − t) = g(x) − h(t). Consequently, the second integral at the right side of (9.24) becomes b K2 (x, t)u(t)dt = αg(x), (9.27) a

and

b

K2 (x, t)u(t)dt = βg(x) − α,

(9.28)

a

by using (9.25) and (9.26) respectively, where b α= h(t)u(t)dt, β = a

b

u(t)dt.

(9.29)

a

The correction functional for the Volterra-Fredholm integro-diﬀerential equation (9.24) is un+1 (x) = un(x) t x (9.30) (i) λ(ξ) un (ξ) − f (ξ) − K1 (ξ, r)˜ un (r)dr − αg(ξ) dξ, + 0

or

0

un+1 (x) = un (x) x t (9.31) (i) + λ(ξ) un (ξ) − f (ξ) − K1 (ξ, r)˜ un (r)dr − βg(ξ) + α dξ, 0

0

by using (9.27) and (9.28) respectively. The variational iteration method is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ(ξ) that can be identiﬁed optimally via integration by parts and by using a restricted variation. Having λ(ξ) determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), n 0 of the solution u(x). The zeroth approximation u0 can be any selective function. However, the given initial values u(0), u (0), . . . are preferably used for selecting the zeroth approximation u0 as will be seen later. Consequently, the solution is given by u(x) = lim un(x). (9.32) n→∞

It is worth noting to summarize the Lagrange multipliers λ(ξ) for a variety of ODEs as formally derived in Chapter 3: u + f (u(ξ), u (ξ)) = 0, λ = −1, u + f (u(ξ), u (ξ), u (ξ)) = 0, λ = ξ − x, 1 (ξ − x)2 , 2! 1 u(iv) + f (u(ξ), u (ξ), u (ξ), u (ξ), u(iv) (ξ)) = 0, λ = (ξ − x)3 , 3! u + f (u(ξ), u (ξ), u (ξ), u (ξ)) = 0, λ = −

9.2 The Volterra-Fredholm Integro-Diﬀerential Equation

291

1 (ξ − x)(n−1) . (n − 1)! (9.33) The VIM will be illustrated by studying the following examples. u(n) + f (u(ξ), u (ξ), u (ξ), . . . , u(n) (ξ)) = 0, λ = (−1)n

Example 9.5 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the variational iteration method x 1 u (x) = 1 + (x − t)u(t)dt + xtu(t), u(0) = 1. (9.34) 0

0

The correction functional for this equation is given by x t (t − r)un (r) dr − αt dt, un (t) − 1 − un+1 (x) = un (x) − 0

(9.35)

0

where we used λ = −1 for ﬁrst-order integro-diﬀerential equation, and 1 α= tu(t)dt. (9.36) 0

We can use the initial condition to select u0 (x) = u(0) = 1. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1, x t (t − r)u0 (r)dr − αt dt u1 (x) = u0 (x) − u0 (t) − 1 − 0

0

0

0

0

0

0

0

1 1 = 1 + x + αx2 + x3 , 2 3! x t u2 (x) = u1 (x) − (t − r)u1 (r)dr − αt dt u1 (t) − 1 − 1 1 1 1 1 = 1 + x + αx2 + x3 + x4 + αx5 + x6 , 2 3! 4! 5! 6! x t u3 (x) = u2 (x) − (t − r)u2 (r)dr − αt dt u2 (t) − 1 −

(9.37)

1 1 1 1 1 = 1 + x + αx2 + x3 + x4 + αx5 + x6 2 3! 4! 5! 6! 1 7 1 8 1 9 + x + αx + x , 7! 8! 9! x t u4 (x) = u3 (x) − (t − r)u3 (r)dr − αt dt u3 (t) − 1 − 1 1 1 1 1 = 1 + x + αx2 + x3 + x4 + αx5 + x6 2 3! 4! 5! 6! 1 1 1 1 10 1 1 12 x + αx11 + x , + x7 + αx8 + x9 + 7! 8! 9! 10! 11! 12! and so on. To determine α, we substitute u4 (x) into (9.36) to ﬁnd that α = 1.

(9.38)

292

9 Volterra-Fredholm Integro-Diﬀerential Equations

Substituting α = 1 into u4 (x), and using u(x) = lim un(x), n→∞

we ﬁnd that the exact solution is u(x) = ex ,

(9.39) (9.40)

x

obtained upon using the Taylor series for e . Example 9.6 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the variational iteration method x 1 (x − t)u(t)dt + (x − t)u(t), u(0) = 2, (9.41) u (x) = 9 − 5x − x2 − x3 + 0

0

that can be written as

2

3

u (x) = 9 − 5x − x − x + where

α=

0

x

(x − t)u(t)dt + αx − β, u(0) = 2,

1

u(t)dt,

1

β=

0

(9.42)

tu(t)dt.

(9.43)

0

The correction functional for the Volterra-Fredholm integro-diﬀerential equation is given by un+1 (x) = un (x) t x (9.44) 2 3 (t − r)un (r)dr − αt + β dt, un (t) − 9 + 5t + t + t − − 0

0

where we used λ = −1 for ﬁrst-order integro-diﬀerential equations. We can use the initial condition to select u0 (x) = u(0) = 2. Using this selection we obtain u0 (x) = 2, u1 (x) = u0 (x) t x 2 3 (t − r)u0 (r)dr − αt + β dt u0 (t) − 9 + 5t + t + t − − 0

0

5−α 2 1 4 = 2 + (9 − β)x − x − x , 2 4 u2 (x) = u1 (x) x t 2 3 − u1 (t) − 9 + 5t + t + t − (t − r)u1 (r)dr − αt + β dt 0

= 2 + (9 − β)x −

0

1 7 5−α 2 3−β 4 5−α 5 x + x − x − x , 2! 4! 5! 840

u3 (x) = u2 (x) t x 2 3 u2 (t) − 9 + 5t + t + t − (t − r)u2 (r)dr − αt + β dt, − 0

5−α 2 3−β 4 x + x = 2 + (9 − β)x − 2! 4!

0

9.2 The Volterra-Fredholm Integro-Diﬀerential Equation

293

5−α 5 3−β 7 5−α 8 (9.45) x + x − x + ··· , 5! 7! 8! and so on. To determine α and β, we substitute u3 (x) into (9.43), and solve the resulting equations to ﬁnd that α = 5, β = 3. (9.46) This in turn gives the exact solution −

u(x) = 2 + 6x, obtained upon substituting α = 5, and β = 3 into un (x).

(9.47)

Example 9.7 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the variational iteration method x 1 2 3 u(t)dt+ (1−2xt)u(t), u(0) = 2, u (0) = 6, u (x) = −8+6x−3x +x + 0

that can be written as

2

3

u (x) = −8+6x−3x +x + where

−1

x

u(t)dt+α−βx, u(0) = 2, u (0) = 6, (9.49)

0

1

u(t)dt,

α= 0

(9.48)

β=

1

2tu(t)dt.

(9.50)

0

The correction functional for the Volterra-Fredholm integro-diﬀerential equation is given by un+1 (x) = un(x) t x 2 3 (t − x) un (t) + 8 − 6t + 3t − t − un (r)dr − α + βt dt, + 0

0

(9.51) where we used λ = (t − x) for second-order integro-diﬀerential equations. We can select u0 (x) = u(0) + u (0)x = 2 + 6x to determine the successive approximations u0 (x) = 2 + 6x,

1 4 1 1 α − 4 x2 + − β x3 + x5 , 2 3 6 20 1 1 4 1 u2 (x) = 2 + 6x + α − 4 x2 + − β x3 + α− 2 3 6 120 1 1 8 1 + − β x6 + x , 90 720 6720 1 4 1 1 u3 (x) = 2 + 6x + α − 4 x2 + − β x3 + α− 2 3 6 120 1 1 1 1 − β x6 + α− x8 + 90 720 40320 20160 u1 (x) = 2 + 6x +

1 60

x5 (9.52)

1 60

x5

294

9 Volterra-Fredholm Integro-Diﬀerential Equations

+

1 1 − β x9 + · · · , 45360 362880

and so on. To determine α and β, we substitute u3 (x) into (9.50), and solve the resulting equations to ﬁnd that α = 2, β = 8. (9.53) This in turn gives the exact solution u(x) = 2 + 6x − 3x2 .

(9.54)

Example 9.8 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the variational iteration x method π 1 2 u (x) = − x + u(t)dt + xu(t), u(0) = u (0) = −u (0) = 1, (9.55) 2 0 −π that can be written as x 1 u (x) = − x2 + u(t)dt + αx, u(0) = u (0) = −u (0) = 1, (9.56) 2 0 where π α= u(t)dt. (9.57) −π

The correction functional for the Volterra-Fredholm integro-diﬀerential equation is given by t 1 x 1 2 2 (t − x) un (t) + t − un (r) dr − αt dt, un+1 (x) = un (x) − 2 0 2 0 (9.58) where we used 1 (9.59) λ = − (t − x)2 , 2 for third-order integro-diﬀerential equations. We can use the initial conditions to select 1 u0 (x) = u(0) + u (0)x + u (0)x2 , 2 1 (9.60) = 1 + x − x2 . 2 Using this selection into the correction functional gives the following successive approximations 1 u0 (x) = 1 + x − x2 , 2! 1 1 1 u1 (x) = 1 + x − x2 + (1 + α)x4 − x6 , 2 4! 6! 1 2 1 1 1 1 10 4 u2 (x) = 1 + x − x + (1 + α)x − x6 + (1 + α)x8 − x , 2 4! 6! 8! 10!

9.2 The Volterra-Fredholm Integro-Diﬀerential Equation

295

1 1 1 u3 (x) = 1 + x − x2 + (1 + α)x4 − x6 2 4! 6! 1 1 10 1 1 14 (1 + α)x8 − x + (1 + α)x12 − x , (9.61) 8! 10! 12! 14! where other approximations are obtained up to u8 (x), but not listed. To determine α, we substitute u8 (x) into (9.57), and solve the resulting equations to ﬁnd that α = 0. (9.62) +

This in turn gives the series solution 1 2 1 4 1 6 1 8 1 10 x + ··· , u(x) = x + 1 − x + x − x + x + 2 4! 6! 8! 10! that converges to the exact solution u(x) = x + cos x.

(9.63)

(9.64)

Exercises 9.2.2 Solve the following Volterra-Fredholm integro-diﬀerential equations by using the variational iteration method x 1 1. u (x) = 6 + 4x − x3 + (x − t)u(t)dt + (1 − xt)u(t)dt, u(0) = 0 0

1 2. u (x) = 6 − 2x − x2 − x3 + 2 3. u (x) = −4 + 6x − x2 − x4 + 4. u (x) = 5x −

1 3 3 x + x5 + 2 4

5. u (x) = −8 −

−1

x 0

0

x

0 x

(x − t)u(t)dt + xu(t)dt +

1

−1

−1

x

0

tu(t)dt +

xu(t)dt, u(0) = 1

(1 − xt)u(t)dt, u(0) = 1

(1 − xt)u(t)dt +

1 2 3 x + x3 + x4 + 2 4

1

1

−1 1

−1

xtu(t)dt, u(0) = 1

(x − t)u(t)dt,

u(0) = 1, u (0) = −3

x 1 2 1 xu(t)dt + (x − t)u(t)dt, u(0) = 3, u (0) = 0 − 3x2 − x5 + 5 4 0 0 x π 1 7. u (x) = −x − x3 + (x − t)u(t)dt + xu(t)dt, u(0) = 0, u (0) = 2 6 0 −π x π (x − t)u(t)dt + xu(t)dt, u(0) = 1, u (0) = 1 8. u (x) = −1 − x +

6. u (x) =

0

x

−π

1

1 1 u(t)dt + tu(t)dt, u(0) = 1, u (0) = 2, u (0) = 1 − x2 + 3 2 0 0 x π 1 10. u (x) = 5 − x4 + u(t)dt + xu(t)dt, u(0) = 0, u (0) = 1, u (0) = 0 4 0 −π x π 11. u (x) = 3 − 4 cos x + u(t)dt + u(t)dt, u(0) = 0, u (0) = 1, u (0) = 0 9. u (x) = −

0

0

296

9 Volterra-Fredholm Integro-Diﬀerential Equations

12. u (x) = −1 − 4 sin x +

x

0

u(t)dt +

π 2

0

u(t)dt, u(0) = 0, u (0) = 0, u (0) = 2

9.3 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations In a parallel manner to our analysis in the previous section, we will study the mixed Volterra-Fredholm integro-diﬀerential equation of the form x b u(i) (x) = f (x) + λ K(r, t)u(t)dtdr, (9.65) 0

a i

where f (x), K(x, t) are analytic functions, and u(i) (x) = ddxui . It is interesting to note that (9.65) contains mixed Volterra and Fredholm integral equations, where the Fredholm integral is the interior integral, whereas the Volterra integral is the exterior one. Moreover, the unknown function u(x) appears inside the integral, whereas the derivative u(i) (x) appears outside the integral. This type of equations will be handled by using the direct computation method and the series solution method. Other methods exist in the literature but will not be presented in this text.

9.3.1 The Direct Computation Method The standard ith order mixed Volterra-Fredholm integro-diﬀerential equation is of the form x

u(i) (x) = f (x) +

b

K(r, t)u(t)dtdr, 0

(9.66)

a i

where f (x), K(x, t) are analytic functions, and u(i) (x) = ddxui . The initial conditions should be prescribed. The unknown function u(x) appears inside the integral, whereas the ith derivative u(i) (x) appears outside the integral. The direct computation method [6] was used before to handle Fredholm integral equation in Chapter 4. In this section, the direct computation method will be used to solve the mixed Volterra-Fredholm integro-diﬀerential equations. The method gives the solution in an exact form and not in a series form. It is important to point out that this method will be applied for the degenerate or separable kernels of the form n K(x, t) = gk (x)hk (t). (9.67) k=1

Examples of separable kernels are x − t, xt, x2 − t2 , xt2 + x2 t, etc. We will focus our study on K(x, t) being separable of the form K(r, t) = g(r)h(t),

(9.68)

9.3 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations

297

or being diﬀerence kernel of the form K(r, t) = K(r − t) = g(r) − h(t). (9.69) Substituting (9.68) or (9.69) into (9.66), the integro-diﬀerential equation becomes x u(i) (x) = f (x) +

βg(r) dr,

(9.70)

(αg(r) − β) dr,

(9.71)

0

or u(i) (x) = f (x) +

x 0

respectively, where

b

α=

u(t)dt,

b

β=

a

h(t)u(t)dt.

(9.72)

a

Integrating (9.70) or (9.71) i times for 0 to x gives the unknown solution u(x), where the constants α and β are to be determined. This can be achieved by substituting the resulting value of u(x) into (9.72). Using the obtained numerical values of α and β, the solution u(x) of the mixed Volterra-Fredholm integro-diﬀerential equation (9.70) or (9.71) is readily obtained. The direct computation method will be illustrated by studying the following mixed Volterra-Fredholm integro-diﬀerential examples. Example 9.9 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the direct computation method x 1 1 ru(t)dtdr, u(0) = 0. (9.73) u (x) = ex (1 + x) − x2 + 2 0 0 This equation can be written as x 1 u (x) = ex (1 + x) − x2 + αrdr, u(0) = 0. (9.74) 2 0 where we used 1

α=

u(t)dt.

(9.75)

0

Integrating both sides once from 0 to x, and using the initial condition we ﬁnd α−1 3 x . u(x) = xex + (9.76) 6 To determine α, we substitute u(x) from (9.76) into (9.75) to ﬁnd that α = 1. (9.77) This in turn gives the exact solution by u(x) = xex .

(9.78)

298

9 Volterra-Fredholm Integro-Diﬀerential Equations

Example 9.10 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the direct computation method x 1 7 2 u (x) = 6 + 29x − x + (r − t)u(t)dtdr, u(0) = 0. (9.79) 2 0 0 This equation can be written as x 7 2 (αr − β)dr, (9.80) u (x) = 6 + 29x − x + 2 0 where 1

1

u(t)dt,

α=

β=

0

tu(t)dt.

(9.81)

0

Integrating both sides once from 0 to x, and using the initial condition we ﬁnd 29 − β 2 α − 1 3 x + x . u(x) = 6x + (9.82) 2 6 To determine α and β, we substitute u(x) from (9.82) into (9.81) to ﬁnd that α = 1, β = 5. (9.83) Substituting α = 7 and β = 5 into u(x) gives the exact solution by u(x) = 6x + 12x2 .

(9.84)

Example 9.11 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the direct computation method x π ru(t)dt dr, u(0) = 1, u (0) = 1. (9.85) u (x) = −x2 − sin x − cos x + 0

0

This equation can be written as u (x) = −x2 − sin x − cos x +

where

x

αr dr,

(9.86)

0

π

u(t)dt.

α=

(9.87)

0

Integrating both sides from 0 to x twice, and using the initial conditions we obtain α−2 4 x . u(x) = sin x + cos x + (9.88) 24 To determine α, we substitute u(x) into (9.87) to ﬁnd that α = 2. Substituting α = 2 into u(x) gives the exact solution by u(x) = sin x + cos x.

(9.89) (9.90)

9.3 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations

299

Example 9.12 Solve the following Volterra-Fredholm integro-diﬀerential equation by using the direct computation method x 1 10 3 u (x) = − + 2x + (rt2 − r2 t)u(t)dt dr, u(0) = 1, u (0) = 9. (9.91) 3 0 −1 Proceeding as before, we set x 10 3 (αr − βr2 ) dr, (9.92) u (x) = − + 2x + 3 0 where 1

α=

t2 u(t)dt,

1

β=

−1

tu(t)dt.

(9.93)

−1

Integrating both sides from 0 to x twice, and using the initial conditions we obtain 5 α 6−β 5 x . u(x) = 1 + 9x − x2 + x4 + (9.94) 3 24 60 To determine α and β, we substitute u(x) into (9.93) to ﬁnd that α = 0, β = 6. Substituting α = 0 and β = 6 into u(x)gives the exact solution by 5 u(x) = 1 + 9x − x2 . 3

(9.95)

(9.96)

Exercises 9.3.1 Solve the following Volterra-Fredholm integro-diﬀerential equations by using the direct computation method x 1 5 1. u (x) = 8x + x2 + (1 − rt)u(t)dtdr, u(0) = 2 4 0 0 x 1 8 2 x + rt2 u(t)dtdr, u(0) = 1 2. u (x) = 1 + 2x − 15 0 −1 x 1 43 2 x + 3. u (x) = 1 − 2x − (1 − rt)u(t)dtdr, u(0) = 1 15 0 −1 x 1 4 3 x + 4. u (x) = 1 − 3x2 − r2 tu(t)dtdr, u(0) = 1 45 0 0 x π 2 5. u (x) = 1 − 2x − sin x + u(t)dtdr, u(0) = 1

0

x

−π 2

1 4 tu(t)dtdr, u(0) = 1 6. u (x) = 1 − x + ex + 3 0 0 x 1 1 7. u (x) = x2 − ex + rtu(t)dtdr, u(0) = 0 4 0 0 x π u(t)dtdr, u(0) = −1 8. u (x) = sin x + cos x + 0

−π

300

9 Volterra-Fredholm Integro-Diﬀerential Equations

9. u (x) = 2x3 −

1 2 x + 3

x

0

1 −1

10. u (x) = −2 sin x − x cos x + 11. u (x) = ex − 3x +

x 0

12. u (x) = −4 sin 2x +

x

0

x

0

π

−π

u(t)dtdr, u(0) = 0, u (0) = 1

tu(t)dtdr, u(0) = 2, u (0) =

0

1

(rt2 − r 2 t)u(t)dtdr, u(0) = 1, u (0) = 9

π

0

11 2

(1 + r)u(t)dtdr, u(0) = 0, u (0) = 2

9.3.2 The Series Solution Method The series solution method was examined before in this chapter. A real function u(x) is called analytic if it has derivatives of all orders such that the generic form of Taylor series at x = 0 can be written as ∞ an xn . (9.97) u(x) = n=0

In this section we will present the series solution method, that stems mainly from the Taylor series for analytic functions, for solving the mixed VolterraFredholm integro-diﬀerential equations. We will assume that the solution u(x) of the mixed Volterra-Fredholm integro-diﬀerential equation x b (i) K(r, t)u(t)dtdr, (9.98) u (x) = f (x) + 0

a

is analytic, and therefore possesses a Taylor series of the form given in (9.97), where the coeﬃcients an will be determined recurrently. In this method, we usually substitute the Taylor series (9.97) into both sides of (9.98) to obtain ∞ (i) ∞ x b k k (9.99) ak x = T (f (x)) + K(r, t) ak t dt dr, 0

k=0

a

or for simplicity we use 2

(i)

(a0 + a1 x + a2 x + · · · )

k=0

x

b

K(r, t) (a0 + a1 t + · · · ) dt dr,

= T (f (x)) + 0

a

(9.100) where T (f (x)) is the Taylor series for f (x). We integrate the inner then the outer integral in (9.99) or (9.100), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine a system of equations in aj , j 0. Solving this system will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (9.97). The exact solution may be obtained if such an

9.3 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations

301

exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. Example 9.13 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the Taylor series solution method x 1 7 u (x) = 6 + 29x − x2 + ru(t)dtdr, u(0) = 0. (9.101) 2 0 0 Substituting u(x) by the series ∞ an xn , (9.102) u(x) = n=0

into both sides of (9.101) leads to ∞ x 1 ∞ 27 2 n n (r − t) an x = 6 + 29x − x + an t dt dr. (9.103) 2 0 0 n=0 n=0 Using the initial condition gives a0 = 0. Evaluating the integrals at the right side, using few terms from both sides, collecting the coeﬃcients of like powers of x, and equating the coeﬃcients of like powers of x in both sides, we obtain a0 = 0, a1 = 6, a2 = 12, aj = 0, j 3. (9.104) The exact solution is therefore given by u(x) = 6x + 12x2 .

(9.105)

Example 9.14 Solve the Volterra-Fredholm integro-diﬀerential equation by using the series solution method x 1 x u (x) = e − x + tu(t)dtdr, u(0) = 1. (9.106) 0

0

Substituting u(x) by the series u(x) =

∞

an xn ,

(9.107)

n=0

into both sides of (9.106) leads to ∞ n an x = ex − x + n=0

0

x 1

t

0

∞

an t

n

dt dr.

(9.108)

n=0

Using the given initial condition gives a0 = 1. Evaluating the integrals at the right side, using few terms from both sides, collecting the coeﬃcients of like powers of x, and equating the coeﬃcients of like powers of x in both sides, we obtain 1 (9.109) an = , n 0. n!

302

9 Volterra-Fredholm Integro-Diﬀerential Equations

The exact solution is therefore given by u(x) = ex .

(9.110)

Example 9.15 Solve the Volterra-Fredholm integro-diﬀerential equation by using the series solution method x 1 u (x) = −2x − cos x + (1 − rt)u(t)dt dr, u(0) = 1, u (0) = 0. (9.111) 0

0

Proceeding as before, we obtain (−1)n , n 0, (2n)! and zero otherwise. This gives the exact solution by a2n =

u(x) = cos x.

(9.112)

(9.113)

Example 9.16 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the series solution method x 1 20 2 (rt2 −r2 t)u(t)dt dr, u(0) = 2, u (0) = 3. (9.114) u (x) = − + x3 + 3 3 0 −1 Using the initial conditions, and proceeding as before we ﬁnd 10 a0 = 2, a1 = 3, a2 = − , aj = 0, j 4. (9.115) 3 The exact solution is given by 10 (9.116) u(x) = 2 + 3x − x2 . 3 Exercises 9.3.2 Use the series solution method to solve the following mixed Volterra-Fredholm integrodiﬀerential equations x 1 5 1. u (x) = 8x + x2 + (1 − rt)u(t)dtdr, u(0) = 2 4 0 0 x 1 127 2. u (x) = 1 − x+ (1 − rt)u(t)dtdr, u(0) = 1 8 0 0 x 1 43 2 x + (1 − rt)u(t)dtdr, u(0) = 1 3. u (x) = 1 − 2x − 15 0 −1 x 1 77 2 x + 4. u (x) = 1 − 2x − (rt2 − r2 t)u(t)dtdr, u(0) = 1 15 0 −1 x π 2 u(t)dtdr, u(0) = 1 5. u (x) = 1 − 2x − sin x + 0

−π 2

9.4 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations 6. u (x) = ex −

3 x+ 2

x

0

1 2 x − ex + 4

8. u (x) =

1 2 x − e−x + 2

10. u (x) = −

1

x

0

1 2 x + 3

tu(t)dtdr, u(0) = 2

0

7. u (x) =

9. u (x) = 2x3 −

1

rtu(t)dtdr, u(0) = 0

0

x 0

0

−1

x

0

10 2 + x3 + 3 9

303

1 −1

x 0

rtu(t)dtdr, u(0) = 1 (rt2 − r 2 t)u(t)dtdr, u(0) = 1, u (0) = 9 1

−1

(rt2 − r2 t)u(t)dtdr, u(0) = 1, u (0) = 1

x 1 77 2 x + rtu(t)dtdr, u(0) = 1, u (0) = 1 120 0 0 x 1 rtu(t)dtdr, u(0) = 1, u (0) = 0 12. u (x) = −15x + 11. u (x) = 2 + 6x −

0

0

9.4 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations in Two Variables In this section we will study the linear mixed Volterra-Fredholm integrodiﬀerential equations in two variables given by t u (x, t) = f˜(x, t) + F (x, t, r, s)u(r, s)drds, (x, t) ∈ Ω × [0, T ], (9.117) 0

Ω

where u(0, t) = u0 . The functions f˜(x), and F (x, t, r, s) are analytic functions on D = Ω×[0, T ], and Ω is a closed subset of Rn , n = 1, 2, 3. It is interesting to note that (9.117) contains mixed Volterra and Fredholm integral equations, where the Fredholm integral is the interior integral, whereas the Volterra integral is the exterior one. Moreover, the unknown function u(x, t) appears inside the integral, whereas the derivative u (x) appears outside the integral. The mixed Volterra-Fredholm integro-diﬀerential equation (9.117) arises from parabolic boundary value problems, and in various physical and biological models. In the literature, some methods, such as the projection method, time collocation method, the trapezoidal Nystrom method, Adomian method, and other analytical or numerical techniques were used to handle this equation. It was found that these techniques encountered diﬃculties in terms of computational work used, and therefore, approximate solutions were obtained for numerical purposes. In particular, it was found that a complicated term f (x, t) can cause diﬃcult integrations and proliferation of terms in Adomian recursive scheme. To overcome the tedious work of the existing strategies, the modiﬁed decomposition method, combined sometimes with the noise terms phenomenon, will form a useful basis for studying the mixed Volterra-Fredholm integro-

304

9 Volterra-Fredholm Integro-Diﬀerential Equations

diﬀerential equation (9.117). The size of the computational work can be dramatically reduced by using the modiﬁed decomposition method. The noise terms were deﬁned as the identical terms with opposite signs that may appear in the ﬁrst two components of the series solution of u(x, t). The modiﬁed decomposition method and the noise terms phenomenon were presented in details in Chapters 3, 4 and in this chapter.

9.4.1 The Modiﬁed Decomposition Method We ﬁrst integrate both sides of (9.117) from 0 to x to obtain x t F (w, t, r, s)u(r, s) dr ds dw, u(x, t) − u(0, t) = f (x, t) + 0

where

(9.118)

Ω

0

x

f˜(w, t)dw,

f (x, t) =

(9.119)

0

and (x, t) ∈ Ω × [0, T ], u(0, t) = u0 . For mixed integro-diﬀerential equations of higher orders, we can integrate the equation as many times as the order of the given equation. However, in this section we will focus our study only on the ﬁrst order equations. The modiﬁed decomposition method expresses the function f (x, t) as the sum of two partial functions, namely f1 (x, t) and f2 (x, t). In other words, we can set f (x, t) = f1 (x, t) + f2 (x, t). (9.120) To minimize the size of calculations, we identify the zeroth component u0 (x, t) by one part of f (x, t), namely f1 (x, t) or f2 (x, t). The other part of f (x, t) can be added to the component u1 (x, t) among other terms. In other words, the modiﬁed decomposition method [7–8] introduces the modiﬁed recurrence relation u0 (x, t) = f1 (x, t), x t u1 (x, t) = f2 (x, t) + F (w, t, r, s)u0 (r, s)drdsdw, (9.121) 0 0 Ω x t F (w, t, r, s)uk (r, s)drdsdw, k 1. uk+1 (x, t) = 0

0

Ω

If noise terms appear between u0 (x, t) and u1 (x, t), then by canceling these terms from u0 (x, t), the remaining non-canceled terms of u0 (x, t) may give the exact solution. This can be satisﬁed by direct substitution. In what follows, we study some illustrative examples. Example 9.17 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the modiﬁed decomposition method

9.4 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations

u (x, t) = t +

1 3 1 t − xt3 + 18 12

305

t 0

Ω

(x − r)(t − s)u(r, s)drds, u(0, t) = 0, (9.122)

with Ω = [0, 1]. Substituting the decomposition series ∞ u(x, t) = un (x, t),

(9.123)

n=0

into (9.122) gives ∞ 1 1 un (x, t) = t + t3 − xt3 18 12 n=0 ∞ t + (x − r)(t − s) un (r, s) drds. Ω

0

(9.124)

n=0

Integrating both sides from 0 to x and using the initial condition we ﬁnd ∞ 1 1 un (x, t) = xt + xt3 − x2 t3 18 24 n=0 ∞ (9.125) x t + (w − t)(r − s) un (r, s) drdsdw. 0

0

Ω

n=0

We then decompose f (x, t) into two parts as follows: 1 1 f0 (x, t) = xt + xt3 , f1 (x, t) = − x2 t3 . (9.126) 18 24 The modiﬁed decomposition technique admits the use of the recursive relation 1 u0 (x, t) = xt + xt3 , 18 x t 1 2 3 u1 (x, t) = − x t + (w − t)(r − s)u0 (r, s)drdsdw, (9.127) 24 0 0 Ω x t (w − t)(r − s)uk (r, s)drdsdw, k 1. uk+1 (x, t) = 0

0

Ω

This gives u0 (x, t) = xt +

1 3 xt , 18

1 1 1 2 5 u1 (x, t) = − xt3 − xt5 + x t . 18 1080 1440

(9.128)

1 1 The self-canceling noise terms 18 xt3 and − 18 xt3 appear between the components u0 (x, t) and u1 (x, t) respectively. By canceling this term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the equation (9.122), the exact solution u(x, t) = xt, (9.129)

306

9 Volterra-Fredholm Integro-Diﬀerential Equations

is readily obtained. Example 9.18 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the modiﬁed decomposition method t 1 1 u (x, t) = 2x − t2 + t4 + rtu(r, s)drds, u(0, t) = −t2 , (9.130) 4 6 0 Ω with Ω = [0, 1]. Substituting the decomposition series ∞ un (x, t), (9.131) u(x, t) = n=0

into the last equation gives ∞ ∞ t 1 2 1 4 un (x, t) = 2x − t + t + rt un (r, s) drds. (9.132) 4 6 0 Ω n=0 n=0 Integrating both sides from 0 to x and using the initial condition we ﬁnd ∞ 1 1 un (x, t) = x2 − t2 − xt2 + xt4 4 6 n=0 ∞ (9.133) x t + rt un (r, s) drdsdw. 0

Ω

0

n=0

We decompose f (x, t) into two parts as follows: 1 1 (9.134) f0 (x, t) = x2 − t2 − xt2 , f1 (x, t) = xt4 . 4 6 The modiﬁed decomposition technique admits the use of the recursive relation 1 u0 (x, t) = x2 − t2 − xt2 , 4 x t 1 4 u1 (x, t) = xt + rtu0 (r, s)drdsdw, (9.135) 6 x t 0 0 Ω uk+1 (x, t) = rtuk (r, s)drdsdw, k 1. 0

0

Ω

This gives 1 u0 (x, t) = x2 − t2 − xt2 , 4 1 2 1 4 u1 (x, t) = xt − xt . 4 36

(9.136)

The self-canceling noise terms − 41 xt2 and 14 xt2 appear between the components u0 (x, t) and u1 (x, t) respectively. By canceling this term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the given equation, the exact solution is given by u(x, t) = x2 − t2 .

(9.137)

9.4 The Mixed Volterra-Fredholm Integro-Diﬀerential Equations

307

Example 9.19 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the modiﬁed decomposition method t 1 1 t 1 2 x u (x, t) = e − + e (1 + t) − t + rsu(r, s)drds, u(0, t) = 1 + et , 2 2 2 Ω 0 (9.138) with Ω = [0, 1]. Proceeding as before, we ﬁnd ∞ 1 1 1 un (x, t) = ex + et − x + xet (1 + t) − xt2 2 2 2 n=0 ∞ (9.139) x t + rs un (r, s) drdsdw. 0

0

Ω

n=0

We decompose f (x, t) into two parts as follows: 1 f0 (x, t) = ex + et − x, 2 (9.140) 1 t 1 f1 (x, t) = xe (1 + t) − xt2 . 2 2 The modiﬁed decomposition technique admits the use of the recursive relation 1 u0 (x, t) = ex + et − x, 2 x t 1 1 u1 (x, t) = xet (1 + t) − xt2 + rsu0 (r, s)drdsdw, (9.141) 2 2 0 0 Ω x t rsuk (r, s)drdsdw, k 1. uk+1 (x, t) = 0

0

Ω

This in turn gives 1 u0 (x, t) = ex + et − x, 2 1 2 1 u1 (x, t) = x − xt . 2 12

(9.142)

The self-canceling noise terms − 21 x and 12 x appear between the components u0 (x, t) and u1 (x, t) respectively. By canceling this term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the given equation, the exact solution is therefore given by u(x, t) = ex + et .

(9.143)

Example 9.20 Solve the mixed Volterra-Fredholm integro-diﬀerential equation by using the modiﬁed decomposition method t 1 3 1 3 u (x, t) = t cos x− t − xt + (xt+rs)u(r, s)drds, u(0, t) = 0, (9.144) 3 2 Ω 0 with Ω = [0, π2 ]. Proceeding as before, we ﬁnd

308

9 Volterra-Fredholm Integro-Diﬀerential Equations ∞

1 1 un (x, t) = t sin x − xt3 − x2 t3 3 4 n=0 ∞ x t + (wt + rs) un (r, s) drdsdw. (9.145) 0

0

Ω

n=0

We decompose f (x, t) into two parts as follows: 1 f0 (x, t) = t sin x − xt3 , 3 (9.146) 1 2 3 f1 (x, t) = − x t . 4 The modiﬁed decomposition technique admits the use of the recursive relation 1 u0 (x, t) = t sin x − xt3 , 3 x t 1 u1 (x, t) = − x2 t3 + (wt + rs)u0 (r, s)drdsdw, (9.147) 4 Ω 0 0 x t uk+1 (x, t) = (wt + rs)uk (r, s)drdsdw, k 1. 0

0

Ω

This in turn gives 1 u0 (x, t) = t sin x − xt3 , 3 1 3 u1 (x, t) = xt + · · · . 3

(9.148)

The self-canceling noise terms − 31 xt3 and 13 xt3 appear between the components u0 (x, t) and u1 (x, t) respectively. By canceling this term from u0 (x, t), and showing that the remaining non-canceled term of u0 (x, t) satisﬁes the given equation, the exact solution is therefore given by u(x, t) = t sin x.

(9.149)

Exercises 9.4.1 Use the modiﬁed decomposition method to solve the mixed Volterra-Fredholm integro-diﬀerential equations in two variables t F (x, t, r, s)u(r, s)drds, (x, t) ∈ Ω × [0, T ], (9.150) u (x, t) = f (x, t) + 0

Ω

where f (x, t), F (x, t, r, s), u(0, t) and Ω are given by: 1 2 1 3 t − t , F = rs, u(0, t) = t, Ω = [0, 1] 6 6 1 2 1 4 2. f = 2x − t − t , F = rs, u(0, t) = t2 , Ω = [0, 1] 8 8 2 3 1 3 3. f = 2t − t − t , F = rs, u(0, t) = 1, Ω = [0, 1] 9 4 1. f = 1 −

References

309

5 2 1 3 1 t− t − t , F = r + s, u(0, t) = t, Ω = [0, 1] 4 12 3 7 4 2 3 2 t − xt , F = x + t + r + s, u(0, t) = 0, Ω = [−1, 1] 5. f = 2xt − 18 9 2 5 1 4 2 6. f = 3x + t + xt , F = x + s, u(0, t) = −t3 , Ω = [−1, 1] 5 2 4. f = 2x −

7. f = −t sin x + t2 , F = r + s, u(0, t) = 0, Ω = [0, π] π π 8. f = cos x − t2 , F = rs, u(0, t) = t, Ω = − , 2 2 π π 9. f = − sin x, F = rs, u(0, t) = 1 + sin t, Ω = − , 2 2 1 2 1 3 t − t , F = rs, u(0, t) = 1 + t, Ω = [0, 1] 2 6 1 1 11. f = et + xt − xtet , F = xt, u(0, t) = 0, Ω = [0, 1] 2 2 1 12. f = tex − t3 , F = rs, u(0, t) = t, Ω = [0, 1] 3

10. f = ex −

References 1. K. Maleknejad and M. Hadizadeh, A New computational method for VolterraFredholm integral equations, Comput. Math. Appl., 37 (1999) 1–8. 2. K. Maleknejad, and Y. Mahmoudi, Taylor polynomial solution of high order nonlinear Volterra-Fredholm integro-diﬀerential equations, Appl. Math. Comput., 145 (2003) 641–653. 3. A.M. Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. Comput., 127 (2002) 405–414. 4. S.A. Youseﬁ, A. Lotﬁ and M. Dehghan, He’s variational iteration method for solving nonlinear mixed Volterra-Fredholm integral equations, Comput. Math. Appl., 58 (2009) 2172–2176. 5. R. Kress, Linear Integral Equations, Springer, Berlin, (1999). 6. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 7. G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994). 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).

Chapter 10

Systems of Volterra Integral Equations

10.1 Introduction Systems of integral equations, linear or nonlinear, appear in scientiﬁc applications in engineering, physics, chemistry and populations growth models [1–4]. Studies of systems of integral equations have attracted much concern in applied sciences. The general ideas and the essential features of these systems are of wide applicability. The systems of Volterra integral equations appear in two kinds. For systems of Volterra integral equations of the ﬁrst kind, the unknown functions appear only under the integral sign in the form: x % 1 (x, t)v(t) + · · · dt, K1 (x, t)u(t) + K f1 (x) = 0

f2 (x) =

x

0

% 2 (x, t)v(t) + · · · dt, K2 (x, t)u(t) + K

(10.1)

.. . However, systems of Volterra integral equations of the second kind, the unknown functions appear inside and outside the integral sign of the form: x % 1 (x, t)v(t) + · · · dt, K1 (x, t)u(t) + K u(x) = f1 (x) + 0

v(x) = f2 (x) + .. .

0

x

% 2 (x, t)v(t) + · · · dt, K2 (x, t)u(t) + K

(10.2)

% i (x, t), and the functions fi (x), i = 1, 2, . . . , n are The kernels Ki (x, t) and K given real-valued functions. A variety of analytical and numerical methods are used to handle systems of Volterra integral equations. The existing techniques encountered some diﬃculties in terms of the size of computational work, especially when the system A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

312

10 Systems of Volterra Integral Equations

involves several integral equations. To avoid the diﬃculties that usually arise from the traditional methods, we will use some of the methods presented in this text. The Adomian decomposition method, the Variational iteration method, and the Laplace transform method will form a reasonable basis for studying systems of integral equations. The emphasis in this text will be on the use of these methods rather than proving theoretical concepts of convergence and existence that can be found in other texts.

10.2 Systems of Volterra Integral Equations of the Second Kind We will ﬁrst study systems of Volterra integral equations of the second kind given by x % 1 (x, t)v(t) + · · · dt, u(x) = f1 (x) + K1 (x, t)u(t) + K 0 (10.3) x % 2 (x, t)v(t) + · · · dt. v(x) = f2 (x) + K2 (x, t)u(t) + K 0

The unknown functions u(x), v(x), . . ., that will be determined, appear in% i (x, t), and the side and outside the integral sign. The kernels Ki (x, t) and K function fi (x) are given real-valued functions. In what follows we will present the methods, new and traditional, that will be used to handle these systems.

10.2.1 The Adomian Decomposition Method The Adomian decomposition method [5–7] was presented before. The method decomposes each solution as an inﬁnite sum of components, where these components are determined recurrently. This method can be used in its standard form, or combined with the noise terms phenomenon. Moreover, the modiﬁed decomposition method will be used wherever it is appropriate. It is interesting to point out that the VIM method can also be used, but we need to transform the system of integral equations to a system of integro-diﬀerential equations that will be presented later in this chapter. Example 10.1 Use the Adomian decomposition method to solve the following system of Volterra integral equations

10.2 Systems of Volterra Integral Equations of the Second Kind

1 u(x) = x − x4 + 6

x

313

(x − t)2 u(t) + (x − t)v(t) dt,

0

(10.4) x

1 v(x) = x2 − x5 + (x − t)3 u(t) + (x − t)2 v(t) dt. 12 0 The Adomian decomposition method suggests that the linear terms u(x) and v(x) be decomposed by an inﬁnite series of components ∞ ∞ un (x), v(x) = vn (x), u(x) = (10.5) n=0

n=0

where un (x) and vn (x), n 0 are the components of u(x) and v(x) that will be elegantly determined in a recursive manner. Substituting (10.5) into (10.4) gives x ∞ ∞ ∞ 1 4 2 un (x) = x − x + un (t) + (x − t) vn (t) dt, (x − t) 6 0 n=0 n=0 n=0 x ∞ ∞ ∞ 1 5 2 3 2 (x − t) vn (x) = x − x + un (t) + (x − t) vn (t) dt. 12 0 n=0 n=0 n=0 (10.6) The zeroth components u0 (x) and v0 (x) are deﬁned by all terms that are not included under the integral sign. Following Adomian analysis, the system (10.6) is transformed into a set of recursive relations given by 1 u0 (x) = x − x4 , 6 x (10.7)

(x − t)2 uk (t) + (x − t)vk (t) dt, k 0, uk+1 (x) = 0

and 1 v0 (x) = x2 − x5 , 12 x

(x − t)3 uk (t) + (x − t)2 vk (t) dt, vk+1 (x) = 0

(10.8) k 0.

This in turn gives 1 u0 (x) = x − x4 , 6 and v0 (x) = x2 −

1 5 x , 12

u1 (x) = v1 (x) =

1 4 1 7 x − x , 6 280 1 5 11 8 x − x . 12 10080

(10.9) (10.10)

It is obvious that the noise terms ∓ 16 x4 appear between u0 (x) and u1 (x). 1 5 Moreover, the noise terms ∓ 12 x appear between v0 (x) and v1 (x). By canceling these noise terms from u0 (x) and v0 (x), the non-canceled terms of u0 (x) and v0 (x) give the exact solutions (10.11) (u(x), v(x)) = (x, x2 ).

314

10 Systems of Volterra Integral Equations

Example 10.2 Use the Adomian decomposition method to solve the following system of Volterra integral equations x (sin(x − t)u(t) + cos(x − t)v(t)) dt, u(x) = cos x − x sin x + 0 (10.12) x v(x) = sin x − x cos x + (cos(x − t)u(t) − sin(x − t)v(t)) dt. 0

We ﬁrst decompose the linear terms u(x) and v(x) by an inﬁnite series of components ∞ ∞ un (x), v(x) = vn (x), u(x) = (10.13) n=0

n=0

where un (x) and vn (x), n 0 are the components of u(x) and v(x) that will be elegantly determined in a recursive manner. Substituting (10.13) into (10.12) gives ∞ un (x) = cos x − x sin x n=0 x ∞ ∞ + un(t) + cos(x − t) vn (t) dt, sin(x − t) 0

∞ n=0

vn (x) = sin x − x cos x x

+ 0

cos(x − t)

n=0

n=0

∞

∞

un (t) − sin(x − t)

n=0

(10.14) vn (t) dt.

n=0

The zeroth components u0 (x) and v0 (x) are deﬁned by all terms that are not included under the integral sign. For this example, we will use the modiﬁed decomposition method, therefore we set the recursive relation u0 (x) = cos x, x (sin(x − t)uk (t) + cos(x − t)vk (t)) dt, k 0, uk+1 (x) = −x sin x + 0

(10.15)

and v0 (x) = sin x, vk+1 (x) = −x cos x + This in turn gives u0 (x) = cos x,

x

0

(cos(x − t)uk (t) − sin(x − t)vk (t)) dt,

k 0. (10.16)

u1 (x) = 0,

uk+1 (x) = 0,

k 1,

(10.17)

v1 (x) = 0,

vk+1 (x) = 0,

k 1.

(10.18)

and v0 (x) = sin x,

10.2 Systems of Volterra Integral Equations of the Second Kind

315

This gives the exact solutions (u(x), v(x)) = (cos x, sin x), that satisfy the system (10.12).

(10.19)

Example 10.3 Use the Adomian decomposition method to solve the following system of Volterra integral equations x

1 1 u(x) = 1 + x2 − x3 − x4 + (x − t)3 u(t) + (x − t)2 v(t) dt, 3 3 0 x

1 1 v(x) = 1 + x − x3 − x4 − x5 + (x − t)4 u(t) + (x − t)3 v(t) dt. 4 4 0 (10.20) Proceeding as before we ﬁnd ∞ 1 1 un (x) = 1 + x2 − x3 − x4 3 3 n=0 x ∞ ∞ 3 2 + un (t) + (x − t) vn (t) dt, (x − t) 0

∞

n=0

n=0

1 1 vn (x) = 1 + x − x3 − x4 − x5 4 4 n=0 x ∞ ∞ 4 3 (x − t) + un (t) + (x − t) vn (t) dt. 0

n=0

(10.21)

n=0

We next set the recursive relations 1 1 u0 (x) = 1 + x2 − x3 − x4 , 3 3 x

uk+1 (x) = (x − t)3 uk (t) + (x − t)2 vk (t) dt, 0

(10.22) k 0,

and 1 1 v0 (x) = 1 + x − x3 − x4 − x5 , 4 4 x

(x − t)4 uk (t) + (x − t)3 vk (t) dt, vk+1 (x) = 0

(10.23) k 0.

This in turn gives

and

1 1 u0 (x) = 1 + x2 − x3 − x4 , 3 3 1 3 1 4 u1 (x) = x + x + · · · , 3 3

(10.24)

1 1 v0 (x) = 1 + x − x3 − x4 − x5 , 4 4 1 1 v1 (x) = x4 + x5 + · · · . 4 4

(10.25)

316

10 Systems of Volterra Integral Equations

It is obvious that the noise terms ∓ 31 x3 and ∓ 31 x4 appear between u0 (x) and u1 (x). Moreover, the noise terms ∓ 14 x4 and ∓ 41 x5 appear between v0 (x) and v1 (x). By canceling these noise terms from u0 (x) and v0 (x), the non-canceled terms of u0 (x) and v0 (x) give the exact solutions (u(x), v(x)) = (1 + x2 , 1 + x − x3 ), (10.26) that satisfy the given system (10.20). Example 10.4 Use the Adomian decomposition method to solve the following system of Volterra integral equations x −t

x e u(t) + et v(t) dt, u(x) = e − 2x + 0 (10.27) x

t −t −x e u(t) + e v(t) dt. v(x) = e + sinh 2x + 0

Proceeding as before we ﬁnd x ∞ ∞ ∞ −t t x un (x) = e − 2x + un (t) + e vn (t) dt, e 0

n=0 ∞

vn (x) = e−x + sinh 2x +

n=0

n=0

x

et 0

n=0 ∞

un (t) + e−t

n=0

∞

(10.28)

vn (t) dt.

n=0

Proceeding as before we set u0 (x) = ex − 2x, and

v0 (x) = e−x + sinh 2x,

u1 (x) = 2x + · · · ,

(10.29)

v1 (x) = − sinh 2x + · · · .

(10.30)

By canceling the noise terms from u0 (x) and v0 (x), the non-canceled terms of u0 (x) and v0 (x) give the exact solutions (u(x), v(x)) = (ex , e−x ), that satisfy the given system (10.27).

(10.31)

Exercises 10.2.1 Use the Adomian decomposition method to solve the following systems of Volterra integral equations ⎧ x

1 5 ⎪ 2 ⎪ x (x − t)2 u(t) + (x − t)v(t) dt u(x) = x − + ⎪ ⎨ 12 0 1. x ⎪

1 6 ⎪ ⎪ (x − t)3 u(t) + (x − t)2 v(t) dt x + ⎩ v(x) = x3 − 30 0

10.2 Systems of Volterra Integral Equations of the Second Kind ⎧ x ⎪ 2 3 ⎪ − x + (xtu(t) + xtv(t)) dt u(x) = 1 + x + x ⎪ ⎨ 0 2. x ⎪ 1 1 ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎩ v(x) = 1 − x − x2 − x3 − x4 − 3 6 0 ⎧ x ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎪ u(x) = 1 − x2 + x3 + ⎨ 0 3. x ⎪ 1 5 ⎪ ⎪ x + ((x − t)u(t) − (x − t)v(t)) dt ⎩ v(x) = 1 − x3 − 10 0 ⎧ x

1 6 ⎪ ⎪ x + (x − t)2 u(t) − (x − t)2 v(t) dt ⎪ u(x) = x2 + x3 − ⎨ 30 0 4. x ⎪ 1 ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎩ v(x) = x2 − x3 − x4 + 6 0 ⎧ x ⎪ ⎪ (cos(x − t)u(t) + sin(x − t)v(t)) dt ⎪ u(x) = cos x − v + ⎨ 5.

0

⎪ ⎪ ⎪ ⎩ v(x) = sin x − xv +

x 0

(sin(x − t)u(t) + cos(x − t)v(t)) dt

⎧ x ⎪ ⎪ (cos(x − t)u(t) + sin(x − t)v(t)) dt u(x) = sin x − xu + ⎪ ⎨ 0 6. x ⎪ ⎪ ⎪ (sin(x − t)u(t) + cos(x − t)v(t)) dt ⎩ v(x) = cos x − u + 0

⎧ x ⎪ 2 ⎪ x − 2 tan x + x + (u(t) + v(t)) dt u(x) = sec ⎪ ⎨ 0 7. x ⎪ ⎪ ⎪ (u(t) − v(t)) dt ⎩ v(x) = tan2 x − x + 0

⎧ x 1 2 ⎪ 2 ⎪ x − + ((x − t)u(t) + (x − t)v(t)) dt u(x) = sin x ⎪ ⎨ 2 0 8. x ⎪

1 1 ⎪ ⎪ (x − t)2 u(t) − (x − t)2 v(t) dt ⎩ v(x) = cos2 x + x − sin 2x + 2 2 0 ⎧ x −t

⎪ −x ⎪ u(x) = e e u(t) + et v(t) dt − sinh 2x + ⎪ ⎨ 9.

⎪ ⎪ ⎪ ⎩ v(x) = ex − 2x +

0

x 0

t

e u(t) + e−t v(t) dt

⎧ ⎪ x ⎪ + u(x) = 1 − 2x + e ⎪ ⎨

x

(u(t) + v(t)) dt

0

10.

x ⎪ 1 ⎪ ⎪ (u(t) − tv(t)) dt ⎩ v(x) = 1 − x + x2 − (x + 1)ex + 2 0 ⎧ x ⎪ ⎪ u(x) = 1 − 2x + sin x + (u(t) + v(t)) dt ⎪ ⎨

11.

⎪ ⎪ ⎪ ⎩ v(x) = 1 − x2 − sin x +

0

x

0

(tu(t) + tv(t)) dt

317

318

10 Systems of Volterra Integral Equations ⎧ ⎪ 2 ⎪ + cos x + u(x) = x − x ⎪ ⎨

12.

x

0

⎪ 2 ⎪ ⎪ ⎩ v(x) = x − x3 − cos x + 3

(u(t) + v(t)) dt x 0

(tu(t) + tv(t)) dt

10.2.2 The Laplace Transform Method The Laplace transform method is a powerful technique that can be used for solving initial value problems and integral equations as well. The Laplace transform method was presented in Chapter 1, and has been used in other chapters in this text. Before we start applying this method, we summarize some of the concepts presented in Section 1.5. In the convolution theorem for the Laplace transform, it was stated that if the kernel K(x, t) of the integral equation x

u(x) = f (x) + λ

K(x, t)u(t)dt,

(10.32)

0

depends on the diﬀerence x − t, then it is called a diﬀerence kernel. Examples of the diﬀerence kernel are ex−t , cos(x − t), and x − t. The integral equation can thus be expressed as x K(x − t)u(t)dt. (10.33) u(x) = f (x) + λ 0

Consider two functions f1 (x) and f2 (x) that possess the conditions needed for the existence of Laplace transform for each. Let the Laplace transforms for the functions f1 (x) and f2 (x) be given by L{f1 (x)} = F1 (s), L{f2 (x)} = F2 (s). (10.34) The Laplace convolution product of these two functions is deﬁned by x (f1 ∗ f2 )(x) = f1 (x − t)f2 (t)dt, (10.35) 0

or (f2 ∗ f1 )(x) =

0

x

f2 (x − t)f1 (t)dt.

(10.36)

Recall that (f1 ∗ f2 )(x) = (f2 ∗ f1 )(x).

(10.37)

We can easily show that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x f1 (x − t)f2 (t)dt} = F1 (s)F2 (s). (10.38) L{(f1 ∗ f2 )(x)} = L{ 0

Based on this summary, we will examine speciﬁc Volterra integral equations where the kernel is a diﬀerence kernel. Recall that we will apply the Laplace transform method and the inverse of the Laplace transform using Table 1.1

10.2 Systems of Volterra Integral Equations of the Second Kind

319

in Section 1.5. The Laplace transform method for solving systems of Volterra integral equations will be illustrated by studying the following examples. Example 10.5 Solve the system of Volterra integral equations by using the Laplace transform method x

u(x) = 1 − x2 + x3 + 3

v(x) = 1 − x −

1 5 x 10

0

((x − t)u(t) + (x − t)v(t)) dt,

+ 0

(10.39)

x

((x − t)u(t) − (x − t)v(t)) dt.

Notice that the kernels K1 (x − t) = K2 (x − t) = x − t. Taking Laplace transform of both sides of each equation in (10.39) gives U (s) = L{u(x)} = L{1 − x2 + x3 } + L{(x − t) ∗ u(x) + (x − t) ∗ v(x)}, 1 V (s) = L{v(x)} = L 1 − x3 − x5 + L{(x − t) ∗ u(x) − (x − t) ∗ v(x)}. 10 (10.40) This in turn gives 2 1 6 1 1 U (s) = − 3 + 4 + 2 U (s) + 2 V (s), s s s s s (10.41) 6 12 1 1 1 V (s) = − 4 − 6 + 2 U (s) − 2 V (s), s s s s s or equivalently 1 6 1 1 2 1 − 2 U (s) − 2 V (s) = − 3 + 4 , s s s s s (10.42) 1 1 6 1 12 1 + 2 V (s) − 2 U (s) = − 4 − 6 . s s s s s Solving this system of equations for U (s) and V (s) gives 3! 1 U (s) = + 4 , s s (10.43) 3! 1 V (s) = − 4 . s s By taking the inverse Laplace transform of both sides of each equation in (10.43), the exact solutions are given by (10.44) (u(x), v(x)) = (1 + x3 , 1 − x3 ). Example 10.6 Solve the system of Volterra integral equations by using the Laplace transform method x

u(x) = cos x − sin x + v(x) = sin x − x sin x +

0 0

(cos(x − t)u(t) + sin(x − t)v(t)) dt, (10.45)

x

(sin(x − t)u(t) + cos(x − t)v(t)) dt.

320

10 Systems of Volterra Integral Equations

Taking Laplace transform of both sides of each equation in (10.45) gives U (s) = L{u(x)} = L{cos x − sin x} + L{cos(x − t) ∗ u(x) + sin(x − t) ∗ v(x)}, (10.46) V (s) = L{v(x)} = L{sin x − x sin x} + L{sin(x − t) ∗ u(x) + cos(x − t) ∗ v(x)}. This in turn gives s 1 s 1 U (s) = − + U (s) + V (s), 2 2 2 1+s 1+s 1+s 1 + s2 (10.47) 2s 1 s 1 − + U (s) + V (s), V (s) = 1 + s2 (1 + s2 )2 1 + s2 1 + s2 or equivalently 1 s s 1 U (s) − 1− V (s) = − , 1 + s2 1 + s2 1 + s2 1 + s2 (10.48) 1 2s s 1 U (s) = − . 1− V (s) − 1 + s2 1 + s2 1 + s2 (1 + s2 )2 Solving this system of equations for U (s) and V (s) gives s 1 U (s) = , V (s) = . (10.49) 1 + s2 1 + s2 By taking the inverse Laplace transform of both sides of each equation in (10.49), the exact solutions are given by (u(x), v(x)) = (cos x, sin x). (10.50) Example 10.7 Solve the system of Volterra integral equations by using the Laplace transform method u(x) = 2 − e−x +

x

((x − t)u(t) + (x − t)v(t)) dt, x v(x) = 2x − ex + 2e−x + ((x − t)u(t) − (x − t)v(t)) dt. 0

(10.51)

0

Taking Laplace transform of both sides of each equation in (10.51) gives U (s) = L{2 − e−x } + L{(x − t) ∗ u(x) + (x − t) ∗ v(x)}, (10.52) V (s) = L{2x − ex + 2e−x } + L{(x − t) ∗ u(x) − (x − t) ∗ v(x)}. This in turn gives 1 1 1 2 + 2 U (s) + 2 V (s), U (s) = − s s+1 s s (10.53) 2 2 1 1 1 V (s) = 2 − + + U (s) − 2 V (s), s s − 1 s + 1 s2 s or equivalently

10.2 Systems of Volterra Integral Equations of the Second Kind

321

1 2 1 1 , 1 − 2 U (s) − 2 V (s) = − s s s s+1 (10.54) 1 1 2 2 1 1 + 2 V (s) − 2 U (s) = 2 − + . s s s s−1 s+1 Solving this system of equations for U (s) and V (s) gives 1 1 U (s) = , V (s) = . (10.55) s−1 s+1 Taking the inverse Laplace transform of (10.55) gives the exact solutions by (u(x), v(x)) = (ex , e−x ).

(10.56)

Example 10.8 Solve the system of Volterra integral equations by using the Laplace transform method x 1 4 1 5 u(x) = x − x − x + ((x − t)v(t) + (x − t)w(t)) dt, 12 20 0x 1 1 (10.57) v(x) = x2 − x3 − x5 + ((x − t)u(t) + (x − t)w(t)) dt, 6 20 x 0 1 5 ((x − t)u(t) + (x − t)v(t)) dt. w(x) = x3 − x4 + 6 12 0 Taking Laplace transform of both sides of each equation in (10.57) gives 1 1 U (s) = L x − x4 − x5 + L {(x − t) ∗ v(x) + (x − t) ∗ w(x)} , 12 20 1 1 5 2 3 V (s) = L x − x − x + L {(x − t) ∗ u(x) + (x − t) ∗ w(x)} , 6 20 5 3 1 x − x4 + L {(x − t) ∗ u(x) + (x − t) ∗ v(x)} W (s) = L 6 12 (10.58) Proceeding as before we ﬁnd 1 1 1 2 6 U (s) − 2 V (s) − 2 W (s) = 2 − 5 − 6 , s s s s s 1 2 1 6 1 (10.59) V (s) − 2 U (s) − 2 W (s) = 3 − 4 − 6 , s s s s s 1 1 5 10 W (s) − 2 U (s) − 2 V (s) = 4 − 5 . s s s s Solving this system of equations for U (s), V (s) and W (s) gives 2! 3! 1 (10.60) U (s) = 2 , V (s) = 3 , W (s) = 4 . s s s By taking the inverse Laplace transform of both sides of each equation in (10.60), the exact solutions are given by (10.61) (u(x), v(x), w(x)) = (x, x2 , x3 ).

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10 Systems of Volterra Integral Equations

Exercises 10.2.2 Use the Laplace transform method to solve the following systems of Volterra integral equations ⎧ x

1 5 ⎪ 2 − ⎪ x (x − t)2 u(t) + (x − t)v(t) dt u(x) = x + ⎪ ⎨ 12 0 1. x ⎪

1 ⎪ 3 6 ⎪ x + (x − t)3 u(t) + (x − t)2 v(t) dt ⎩ v(x) = x − 30 0 ⎧ x ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎪ u(x) = 1 + x + ⎨ 0

x ⎪ 2 ⎪ ⎪ (u(t) − v(t)) dt ⎩ v(x) = 1 − x − 2x2 − x3 + 3 0 ⎧ x 1 5 ⎪ ⎪ ((x − t)u(t) − (x − t)v(t)) dt x + ⎪ u(x) = 1 + x3 − ⎨ 10 0 3. x ⎪ 1 ⎪ ⎪ (u(t) − v(t)) dt ⎩ v(x) = 1 − x3 − x4 + 2 0 ⎧ x

1 6 ⎪ 2 3 ⎪ (x − t)2 u(t) − (x − t)2 v(t) dt ⎪ ⎨ u(x) = x + x − 30 x + 0 4. x ⎪ 1 ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎩ v(x) = x2 − x3 − x4 + 6 0 ⎧ x ⎪ ⎪ (cos(x − t)u(t) + sin(x − t)v(t)) dt ⎪ ⎨ u(x) = cos x + 2 sin x − 1 + 2.

5.

0

⎪ ⎪ ⎪ ⎩ v(x) = sin x + x cos x − 1 + ⎧ ⎪ ⎪ u(x) = x − x sin x + ⎪ ⎨

6.

x

0 x

⎪ ⎪ ⎪ ⎩ v(x) = 2 − 2 cosh x +

0

(sin(x − t)u(t) + cos(x − t)v(t)) dt

(sin(x − t)u(t) + cos(x − t)v(t)) dt

(cosh(x − t)u(t) + sinh(x − t)v(t)) dt

0

x

(cos(x − t)u(t) + sin(x − t)v(t)) dt

0

⎪ ⎪ ⎪ ⎩ v(x) = cos x − sin x + ⎧ ⎪ ⎪ u(x) = 2 − cosh x + ⎪ ⎨

7.

x

x

0

(sinh(x − t)u(t) + cosh(x − t)v(t)) dt

⎧ x ⎪ ⎪ (cosh(x − t)u(t) + sinh(x − t)v(t)) dt ⎪ u(x) = ex − sinh x − x sinh x + ⎨ 0 8. x ⎪ ⎪ ⎪ (sinh(x − t)u(t) + cosh(x − t)v(t)) dt ⎩ v(x) = e−x − x cosh x + 0

⎧ x ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎪ ⎨ u(x) = cos x + 3 sin x − 2x + 0 9. x ⎪ ⎪ ⎪ ((x − t)u(t) − (x − t)v(t)) dt ⎩ v(x) = cos x + sin x − 2 + 0

10.3 Systems of Volterra Integral Equations of the First Kind ⎧ x 1 x ⎪ ⎪ (sin x + cos x) + ((x − t)u(t) + (x − t)v(t)) dt e u(x) = ⎪ ⎨ 2 0 10. x ⎪ ⎪ ⎪ (u(t) + v(t)) dt ⎩ v(x) = ex (cos x − sin x) +

323

0

⎧ x ⎪ ⎪ (v(t) + w(t)) dt u(x) = cos x − sin x − 1 + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ x ⎨ (u(t) + w(t)) dt v(x) = 3 cos x − sin x − 2 + 11. ⎪ 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (u(t) + v(t)) dt ⎩ w(x) = 2 cos x − 1 + 0

⎧ x 1 4 ⎪ 2 ⎪ x + + (u(t) + v(t) − w(t)) dt u(x) = 1 − x ⎪ ⎪ ⎪ 4 0 ⎪ ⎪ ⎪ x ⎨ 2 1 (v(t) + w(t) − u(t)) dt v(x) = 2x + x2 − x3 − x4 + 12. ⎪ 3 4 0 ⎪ ⎪ ⎪ ⎪ x ⎪ 1 ⎪ ⎪ (w(t) + u(t) − v(t)) dt ⎩ w(x) = −x + x2 + x3 − x4 + 4 0

10.3 Systems of Volterra Integral Equations of the First Kind The standard form of the systems of Volterra integral equations of the ﬁrst kind is given by x % 1 (x, t)v(t) + · · · dt, f1 (x) = K1 (x, t)u(t) + K 0

f2 (x) = .. .

0

x

% 2 (x, t)v(t) + · · · dt, K2 (x, t)u(t) + K

(10.62)

% i (x, t), and the functions fi (x) are given where the kernels Ki (x, t) and K real-valued functions, and u(x), v(x), . . . are the unknown functions that will be determined. Recall that the unknown functions appear inside the integral sign for the Volterra integral equations of the ﬁrst kind. In this section we will discuss two main methods that are commonly used for handling the Volterra integral equations of the ﬁrst kind. Other methods are available in the literature but will not be presented in this text.

10.3.1 The Laplace Transform Method We ﬁrst begin by using the Laplace transform method. Recall that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by

324

10 Systems of Volterra Integral Equations

L{(f1 ∗ f2 )(x)} = L

0

x

f1 (x − t)f2 (t)dt

= F1 (s)F2 (s).

(10.63)

proceeding as in the previous section, we will examine speciﬁc systems of Volterra integral equations where the kernel is a diﬀerence kernel. We will apply the Laplace transform method and the inverse of the Laplace transform using Table 1.1 in Section 1.5. The Laplace transform method for solving systems of Volterra integral equations will be illustrated by studying the following examples. Example 10.9 Solve the system of Volterra integral equations of the ﬁrst kind by using the Laplace transform method x 1 2 1 3 1 4 x + x + x = ((x − t − 1)u(t) + (x − t + 1)v(t)) dt, 2 2 12 0 (10.64) x 1 4 3 2 1 3 x − x + x = ((x − t + 1)u(t) + (x − t − 1)v(t)) dt. 2 6 12 0 Taking Laplace transform of both sides of each equation in (10.64) gives 1 1 2 1 3 x + x + x4 = L {(x − t − 1) ∗ u(x) + (x − t + 1) ∗ v(x)} , L 2 2 12 1 3 2 1 3 x − x + x4 = L {(x − t + 1) ∗ u(x) + (x − t − 1) ∗ v(x)} . L 2 6 12 (10.65) This in turn gives 1 1 1 1 3 2 1 U (s) + V (s) = 3 + 4 + 5 , − + s2 s s2 s s s s (10.66) 1 1 1 2 1 1 3 + − − + U (s) + V (s) = s2 s s2 s s3 s4 s5 Solving this system of equations for U (s) and V (s) gives 1 2 1 1 U (s) = + 2 , V (s) = + 3 . (10.67) s s s s By taking the inverse Laplace transform of both sides of each equation in (10.67), the exact solutions are given by (10.68) (u(x), v(x)) = (1 + x, 1 + x2 ). Example 10.10 Solve the system of Volterra integral equations of the ﬁrst kind by using the Laplace transform method x ((x − t)u(t) + (x − t + 1)v(t)) dt, ex − 1 = 0 (10.69) x e−x − 1 = ((x − t − 1)u(t) + (x − t)v(t)) dt. 0

Taking Laplace transform of both sides of each equation in (10.69) gives

10.3 Systems of Volterra Integral Equations of the First Kind

325

L{ex − 1} = L{(x − t) ∗ u(x) + (x − t + 1) ∗ v(x)}, (10.70) L{e−x − 1} = L{(x − t − 1) ∗ u(x) + (x − t) ∗ v(x)}. This in turn gives 1 1 1 1 1 V (s) = − , U (s) + + 2 2 s s s s−1 s (10.71) 1 1 1 1 1 − V (s) = U (s) + − . s2 s s2 s+1 s Solving this system of equations for U (s) and V (s) gives 1 1 , V (s) = . (10.72) U (s) = s−1 s+1 By taking the inverse Laplace transform of both sides of each equation in (10.72), the exact solutions are given by (u(x), v(x)) = (ex , e−x ). (10.73) Example 10.11 Solve the system of Volterra integral equations of the ﬁrst kind by using the Laplace transform method x ((x − t − 1)u(t) − (x − t)v(t)) dt, 1 − sin x − cos x = 0 (10.74) x ((x − t)u(t) − (x − t − 1)v(t)) dt. 3 − sin x − 3 cos x = 0

Taking Laplace transform of both sides of each equation in (10.74) gives 1 1 1 1 s+1 − U (s) − 2 V (s) = − 2 , 2 s s s s s +1 (10.75) 1 1 3 3s + 1 1 U (s) − − V (s) = − . s2 s2 s s s2 + 1 Solving this system of equations for U (s) and V (s) gives s 1 s 1 + , V (s) = 2 − . (10.76) U (s) = 2 s + 1 s2 + 1 s + 1 s2 + 1 Taking the inverse Laplace transform of (10.76) gives the exact solutions by (u(x), v(x)) = (sin x + cos x, sin x − cos x). (10.77) Example 10.12 Solve the system of Volterra integral equations of the ﬁrst kind by using the Laplace transform method

326

10 Systems of Volterra Integral Equations

−2 + 2 cosh x = 1 −1 + x + x2 + ex = 2

x

(u(t) − v(t)) dt,

0 x

((x − t + 1)v(t) − (x − t − 1)w(t)) dt,

0

(10.78)

x 1 ((x − t)w(t) + (x − t + 1)u(t)) dt. x + x2 + xex = 2 0 Taking Laplace transform of both sides of each equation in (10.78) and solving the system of equations for U (s), V (s) and W (s) we ﬁnd 1 1 s 1 1 , V (s) = + , W (s) = U (s) = + (10.79) s s−1 s s+1 (s − 1)2 Taking the inverse Laplace transform of (10.79) gives the exact solutions by (u(x), v(x), w(x)) = (1 + ex , 1 + e−x , xex ).

(10.80)

Exercises 10.3.1 Use the Laplace transform method to solve the following systems of Volterra integral equations of the ﬁrst kind ⎧ x ⎪ ⎪ x2 + 2 x3 = ((x − t + 1)u(t) + (x − t − 1)v(t)) dt ⎪ ⎨ 3 0 1. x ⎪ 2 2 3 ⎪ ⎪ ((x − t − 1)u(t) + (x − t + 1)v(t)) dt ⎩x − x = 3 0 ⎧ x ⎪ 3 ⎪ ((x − t + 1)u(t) + (x − t − 1)v(t)) dt ⎪ ⎨x = 0

x ⎪ 1 ⎪ ⎪ ((x − t − 1)u(t) + (x − t + 1)v(t)) dt ⎩ − x3 = 3 0 ⎧ x 1 4 ⎪ 2 ⎪ 2x + = ((x − t + 1)u(t) + (x − t − 1)v(t)) dt x ⎪ ⎨ 2 0 3. x ⎪ 1 ⎪ ⎪ ((x − t − 1)u(t) + (x − t + 1)v(t)) dt ⎩ − x4 = 2 0 ⎧ x ⎪ ⎪ 2 + x − sin x − 2 cos x = ((x − t + 1)u(t) + (x − t)v(t)) dt ⎪ ⎨ 2.

4.

⎪ ⎪ ⎪ ⎩ 1 + x − cos x =

0

x 0

((x − t)u(t) + (x − t + 1)v(t)) dt

⎧ x 1 1 ⎪ ⎪ ((x − t + 1)u(t) + (x − t)v(t)) dt ⎪ x + x2 + x3 − sin x = ⎨ 2 3 0 5. x ⎪ ⎪ ⎪ ((x − t − 1)u(t) − (x − t + 1)v(t)) dt ⎩ x − x2 = 0

⎧ x ⎪ 2 ⎪ 2x + x + 2 sin x = ((x − t + 2)u(t) + (x − t − 2)v(t)) dt ⎪ ⎨ 0 6. x ⎪ ⎪ ⎪ ((x − t − 1)u(t) − (x − t + 1)v(t)) dt ⎩ −2x = 0

10.3 Systems of Volterra Integral Equations of the First Kind ⎧ x ⎪ 2 x ⎪ + 4e = ((x − t + 2)u(t) + (x − t − 2)v(t)) dt −4 + x ⎪ ⎨ 0 7. x ⎪ ⎪ ⎪ ((x − t + 1)u(t) − (x − t + 1)v(t)) dt ⎩ −2 − 4x + 2ex =

327

0

⎧ ⎪ x x ⎪ + 4xe = 6 − 6e ⎪ ⎨ 8.

0

⎪ ⎪ ⎪ ⎩ 4 − 4ex + 2xex =

⎧ ⎪ x ⎪ −2 − 2x + 2e = ⎪ ⎨ 9.

x

x

0 x

0

((x − t)u(t) − (x − t + 2)v(t)) dt ((x − t − 1)u(t) − (x − t + 1)v(t)) dt

((x − t − 1)u(t) + (x − t + 1)v(t)) dt

⎪ ⎪ ⎪ ⎩ −2 − 2x + 2x2 + 2ex =

x

0 x

((x − t + 1)u(t) + (x − t − 1)v(t)) dt

⎧ 1 3 1 4 ⎪ 2 ⎪ + + = ((x − t)u(t) + (x − t)v(t)) dt x x x ⎪ ⎪ ⎪ 6 12 0 ⎪ ⎪ ⎪ ⎨ x 1 1 ((x − t)v(t) + w(t)) dt 10. x + x2 + x4 = ⎪ 2 3 0 ⎪ ⎪ ⎪ x ⎪ ⎪ 1 3 ⎪ ⎪ ((x − t)w(t) + u(t)) dt x = ⎩ x + x2 + 20 0 ⎧ x 1 ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt x + x2 − sin x = ⎪ ⎪ ⎪ 2 0 ⎪ ⎪ ⎪ x ⎨ ((x − t)v(t) + w(t)) dt 11. x = ⎪ 0 ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ((x − t)w(t) + u(t)) dt ⎩ 1 + x − cos x = 0

⎧ x ⎪ ⎪ ((x − t)u(t) − (x − t)v(t) + w(t)) dt 2 − x − 2 cos x = ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ x ⎨ 1 (u(t) − (x − t)v(t) + w(t)) dt 12. 1 − x2 + sin x − cos x = ⎪ 2 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (u(t) − v(t) + (x − t)w(t)) dt ⎩ −2 + x + 2 cos x = 0

10.3.2 Conversion to a Volterra System of the Second Kind The conversion technique works eﬀectively by using Leibnitz rule that was presented in Section 1.3. Diﬀerentiating both sides of each equation in (10.62), and using Leibnitz rule, we obtain

328

10 Systems of Volterra Integral Equations

% 1 (x, x)v(x) f1 (x) = K1 (x, x)u(x) + K x % 1x (x, t)v(t) + · · · dt, + K1x (x, t)u(t) + K 0

f2 (x)

% 2 (x, x)v(x) = K2 (x, x)u(x) + K x % 2x (x, t)v(t) + · · · dt, + K2x (x, t)u(t) + K

(10.81)

0

.. . Three remarks can be made here: % i (x, x), i = 1, 2, . . . , n in each of the 1. If at least one of Ki (x, x) and K above equations does not vanish, then the system is reduced to a system of Volterra integral equations of the second kind. In this case, we can use any method that we studied before. % i (x, x) = 0, i = 1, 2, . . . , n, for any equation, and 2. If Ki (x, x) = 0 and K % if Kix (x, x) = 0 and Kix (x, x) = 0, then we diﬀerentiate again that equation. 3. The functions fi (x) must satisfy speciﬁc conditions to guarantee a unique continuous solution for each of the unknown solutions. The determination of these special conditions will be left as an exercise.

10.4 Systems of Volterra Integro-Diﬀerential Equations Volterra studied the hereditary inﬂuences when he was examining a population growth model. The research work resulted in a speciﬁc topic, where both diﬀerential and integral operators appeared together in the same equation. This new type of equations was termed as Volterra integro-diﬀerential equations, given in the form x u(i) (x) = f (x) + K(x, t)u(t)dt, (10.82) 0

n

where u(i) (x) = ddxui . Because the resulted equation combines the diﬀerential operator and the integral operator, then it is necessary to deﬁne initial conditions u(0), u (0), . . . , u(i−1) (0) for the determination of the particular solution u(x) of the Volterra integro-diﬀerential equation. The integro-diﬀerential equations were investigated in Chapter 5. In this section, we will study systems of Volterra integro-diﬀerential equations of the second kind given by x (i) % 1 (x, t)v(t) + · · · dt, u (x) = f1 (x) + K1 (x, t)u(t) + K 0 (10.83) x (i) % K2 (x, t)u(t) + K2 (x, t)v(t) + · · · dt. v (x) = f2 (x) + 0

10.4 Systems of Volterra Integro-Diﬀerential Equations

329

The unknown functions u(x), v(x), . . ., that will be determined, occur inside the integral sign whereas the derivatives of u(x), v(x), . . . appear mostly out% i (x, t), and the function side the integral sign. The kernels Ki (x, t) and K fi (x) are given real-valued functions. There is a variety of numerical and analytical methods that will be used for solving the system of integro-diﬀerential equations. However, in this section, we will present only two methods, new and traditional, that will be used for this study.

10.4.1 The Variational Iteration Method In Chapter 3, the variational iteration method (VIM) was used before in this text. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method [8] handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the so called Adomian polynomials that we need for nonlinear terms. The correction functionals for the Volterra system of integro-diﬀerential equations (10.83) are given by x t λ(t) u(i) (t) − f (t) − K(t, r)˜ u (r)dr dt, un+1 (x) = un (x) + 1 n n vn+1 (x) = vn (x) +

0 x

λ(t)

0

vn(i) (t)

− f2 (t) −

0 t

0

K(t, r)˜ vn (r)dr dt.

(10.84) As presented before, the variational iteration method is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ that can be identiﬁed optimally via integration by parts and by using a restricted variation. Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), n 0 and vn+1 (x), n 0 of the solutions u(x) and v(x). The zeroth approximations u0 (x) and v0 (x) can be any selective functions. However, the initial conditions are preferably used to select these approximations u0 and v0 (x) as will be seen later. Consequently, the solutions are given by u(x) = lim un (x), v(x) = lim vn (x). (10.85) n→∞

n→∞

The VIM will be illustrated by studying the following examples. Example 10.13 Use the VIM to solve the system of Volterra integro-diﬀerential equations

330

10 Systems of Volterra Integral Equations

1 1 u (x) = 1 + x − x2 + x3 + 2 3

x

0

3 1 v (x) = −1 − 3x − x2 − x3 + 2 3

((x − t)u(t) + (x − t + 1)v(t))dt,

x 0

((x − t + 1)u(t) + (x − t)v(t))dt,

(10.86) where u(0) = 1, v(0) = 1. The correction functionals for this system are given by un+1 (x) = un (x) x 1 2 1 3 un (t) − 1 − t + t − t − I1 (t) dt, − 2 3 0 x 3 2 1 3 vn (t) + 1 + 3t + t + t − I2 (t) dt, (10.87) vn+1 (x) = vn (x) − 2 3 0 where t

I1 (t) =

I2 (t) =

((t − r)un (r) + (t − r + 1)vn (r))dr,

0

0

(10.88)

t

((t − r + 1)un (r) + (t − r)vn (r))dr,

and λ = −1 for ﬁrst order integro-diﬀerential equation. We can use the initial conditions to select u0 (x) = u(0) = 1 and v0 (x) = v(0) = 1. Using this selection into the correction functionals gives the following successive approximations

u0 (x) = 1, v0 (x) = 1,

⎧ 1 3 1 4 ⎪ 2 ⎪ x , ⎨ u1 (x) = 1 + x + x + x + 6 12 ⎪ 1 1 4 ⎪ ⎩ v1 (x) = 1 − x − x2 − x3 − x , 6 12 ⎧ ⎪ u2 (x) = 1 + x + x2 + 1 x3 − 1 x3 + 1 x4 − 1 x4 − 1 x5 − 1 x6 , ⎪ ⎨ 6 6 12 12 120 360 ⎪ 1 1 1 1 1 1 6 ⎪ ⎩ v2 (x) = 1 − x − x2 + x3 − x3 + x4 − x4 + x5 + x , 6 6 12 12 120 360 ⎧ 1 6 1 5 1 5 1 6 ⎪ ⎪ u3 (x) = 1 + x + x2 + x − x + x − x + ··· , ⎨ 120 120 360 360 ⎪ 1 6 1 5 1 5 1 6 ⎪ ⎩ v3 (x) = 1 − x − x2 + + x − x x − x + ··· , 120 120 360 360

and so on. By canceling the noise terms, the exact solutions are given by (u(x), v(x)) = (1 + x + x2 , 1 − x − x2 ). (10.89) Example 10.14 Use the VIM to solve the system of Volterra integro-diﬀerential equations

10.4 Systems of Volterra Integro-Diﬀerential Equations

u (x) = 1 − x2 + ex + v (x) = 3 − 3ex +

0

x

(u(t) + v(t))dt, u(0) = 1, v(0) = −1, (10.90)

x

0

331

(u(t) − v(t))dt.

The correction functionals for this system are given by x t un (t) − 1 + t2 − et − un+1 (x) = un (x) − (un (r) + vn (r))dr dt, vn+1 (x) = vn (x) −

0 x

0

0

t (un (r) − vn (r))dr dt. vn (t) − 3 + 3et − 0

(10.91) We use the given initial conditions to select the zeroth approximations u0 (x) = u(0) = 1 and v0 (x) = v(0) = −1. Using this selection into the correction functionals gives the following successive approximations

u0 (x) = 1 v0 (x) = −1

x t ⎧ 2 t ⎪ ⎪ u (t) − 1 + t − e − (u u 1 (x) = u0 (x) − 0 (r) + v0 (r))dr dt ⎪ 0 ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ 1 3 ⎪ ⎨ = 1− x 3 ⎪ x t ⎪ ⎪ ⎪ t ⎪ v v (x) = v (x) − (t) − 3 + 3e − (u (r) − v (r))dr dt ⎪ 1 0 0 0 0 ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎩ = 2 + 3x − 3ex + x2 x t ⎧ 2 t ⎪ ⎪ u (x) = u (x) − (t) − 1 + t − e − (u (r) + v (r))dr dt u 2 1 1 1 1 ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ 1 3 1 4 1 2 ⎪ ⎪ + + + · · · = x + 1 + x + x x x ⎪ ⎨ 2! 3! 4! x t ⎪ ⎪ ⎪ ⎪ (u1 (r) − v1 (r))dr dt v2 (x) = v1 (x) − v1 (t) − 3 + 3et − ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ 1 2 1 3 1 4 1 ⎪ ⎩ = x − 1 + x + x + x + x + x5 + · · · 2! 3! 4! 5! ⎧ 1 1 1 1 ⎪ ⎪ u3 (x) = x + 1 + x + x2 + x3 + x4 + x5 + · · · ⎪ ⎨ 2! 3! 4! 5! ⎪ 1 1 1 1 1 ⎪ ⎪ ⎩ v3 (x) = x − 1 + x + x2 + x3 + x4 + x5 + x6 + · · · 2! 3! 4! 5! 6!

and so on. The exact solutions are therefore given by (u(x), v(x)) = (x + ex , x − ex ).

(10.92)

Example 10.15 Use the VIM to solve the system of Volterra integro-diﬀerential equations

332

10 Systems of Volterra Integral Equations

u (x) = −1 − x2 − sin x +

x

(u(t) + v(t))dt, u(0) = 1, u (0) = 1,

0

v (x) = 1 − 2 sin x − cos x +

x

0

(u(t) − v(t))dt, v(0) = 0, v (0) = 2.

(10.93) The correction functionals for this system are given by un+1 (x) = un (x) t x 2 (un (r) + vn (r))dr) dt, + (t − x)(un (t) + 1 + t + sin t − 0

0

vn+1 (x) = vn (x) x t (t − x)(vn (t) − 1 + 2 sin t + cos t − + (un (r) − vn (r))dr) dt, 0

0

(10.94) where we used λ = t − x for second-order integro-diﬀerential equation. We use the initial conditions to select u0 (x) = 1 + x and v0 (x) = 2x. Using this selection into the correction functionals gives the following successive approximations

u0 (x) = 1 + x,

v0 (x) = 2x, ⎧ 1 2 1 3 1 4 ⎪ ⎪ ⎨ u1 (x) = 1 − x + x + x + sin x, 2! 3! 4! ⎪ 1 1 1 ⎪ ⎩ v1 (x) = cos x + 2 sin x + x2 + x3 − x4 − 1, 2! 3! 4! ⎧ 1 1 1 ⎪ ⎪ u2 (x) = x + 1 − x2 + x4 − x6 + · · · , ⎨ 2! 4! 6! ⎪ 1 1 1 ⎪ ⎩ v2 (x) = x + x − x3 + x5 + x7 + · · · , 3! 5! 7!

and so on. The exact solutions are therefore given by (u(x), v(x)) = (x + cos x, x + sin x).

(10.95)

Example 10.16 Use the variational iteration method to solve the system of Volterra integrodiﬀerential equations x (6v(t) − 3w(t))dt, u(0) = 1, u (x) = 2 + ex − 3e2x + e3x + v (x) = ex + 2e2x − e3x +

2x

w (x) = −e + e x

+ 3e

3x

0

x

0

(3w(t) − u(t))dt, v(0) = 1,

+ 0

x

(u(t) − 2v(t))dt, w(0) = 1.

The correction functionals for this system are given by

(10.96)

10.4 Systems of Volterra Integro-Diﬀerential Equations

333

un+1 (x) = un (x) t x t 2t 3t (6vn (r) − 3wn (r)dr) dt, un (t) − 2 − e + 3e − e − − 0

0

vn+1 (x) = vn (x) t x (3wn (r) − un (r))dr dt, vn (t) − et − 2e2t + e3t − − 0

(10.97)

0

wn+1 (x) = wn (x) t x (un (r) − 2vn (r))dr dt. − wn (t) + et − e2t − 3e3t − 0

0

The initial conditions can be used to select the zeroth approximations as u0 (x) = 1, v0 (x) = 1 and w0 (x) = 1. Using this selection into the correction functionals gives the following successive approximations ⎧ ⎪ ⎨ u0 (x) = 1 v0 (x) = 1 ⎪ ⎩ w0 (x) = 1 ⎧ 7 3 1 3 ⎪ ⎪ u1 (x) = + 2x + x2 + ex − e2x + e3x ⎪ ⎪ 6 2 2 3 ⎪ ⎪ ⎨ 2 1 3x 2 x 2x v1 (x) = − + x + e + e − e ⎪ 3 3 ⎪ ⎪ ⎪ ⎪ 1 1 1 ⎪ ⎩ w1 (x) = − x2 − ex + e2x + e3x 2 2 2 ⎧ 1 1 ⎪ u2 (x) = 1 + x + x2 + x3 + 1 x4 + · · · ⎪ ⎪ ⎪ 2! 3! 4! ⎪ ⎪ ⎨ 1 1 1 2 v2 (x) = 1 + 2x + (2x) + (2x)3 + (2x)4 + · · · ⎪ 2! 3! 4! ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ w2 (x) = 1 + 3x + 1 (3x)2 + 1 (3x)3 + 1 (3x)4 + · · · 2! 3! 4!

and so on. The exact solutions are therefore given by (u(x), v(x), w(x)) = (ex , e2x , e3x ).

(10.98)

Exercises 10.4.1 Use the variational iteration method to solve the following systems of Volterra integrodiﬀerential equations ⎧ x 1 3 ⎪ ⎪ x u (x) = 2x + + ((1 − xt)u(t) − (1 + xt)v(t)) dt, u(0) = 1 ⎪ ⎨ 3 0 1. x ⎪ 1 ⎪ ⎪ ((1 + xt)u(t) + (1 − xt)v(t)) dt, v(0) = 1 ⎩ v (x) = −4x − x5 + 2 0

334 ⎧ ⎪ ⎪ u (x) ⎪ ⎪ ⎪ ⎨ 2. ⎪ ⎪ ⎪ v (x) ⎪ ⎪ ⎩ u(0) ⎧ ⎪ ⎪ ⎪ ⎨ u (x) 3.

10 Systems of Volterra Integral Equations = 3 − 3x +

3 3 x + 2

= −3 − 3x −

x 0

3 3 x + 2

= 1, v(0) = 2 = 1 − 4x + x3 +

⎪ ⎪ ⎪ ⎩ v (x) = −1 − x3 +

((1 − xt)u(t) + (1 − xt)v(t)) dt x

0

x

0

((1 + xt)u(t) + (1 + xt)v(t)) dt

((1 − xt)u(t) + (1 − xt)v(t)) dt, u(0) = 1

x

((1 + xt)u(t) + (1 + xt)v(t)) dt, v(0) = 1

0

x ⎧ ⎪ ⎪ ((1 − xt)u(t) + (1 − xt)v(t)) dt ⎪ u (x) = −2x + x3 + cos x + ⎪ ⎪ 0 ⎨ x 4. ⎪ ((1 + xt)u(t) + (1 + xt)v(t)) dt v (x) = −2x − x3 − cos x + ⎪ ⎪ ⎪ 0 ⎪ ⎩ u(0) = 1, v(0) = 1 ⎧ x ⎪ ⎪ (x) = 1 − x + sin x + ((x − t)u(t) − (x − t)v(t)) dt, u(0) = 1 u ⎪ ⎨ 5.

⎪ ⎪ ⎪ ⎩ v (x) = −1 + cos x +

0

x

0

(u(t) − v(t)) dt, v(0) = 2

x ⎧ ⎪ 2 ⎪ (x) = 1 − x − sin x + (u(t) + v(t)) dt u ⎪ ⎪ ⎪ 0 ⎨ x 6. 2 ⎪ ((1 − xt)u(t) − (1 + xt)v(t)) dt v (x) = 1 + x4 − sin x + ⎪ ⎪ ⎪ 3 0 ⎪ ⎩ u(0) = 1, v(0) = 1 x ⎧ ⎪ 2x ⎪ (x) = 2 − e + (u(t) + v(t)) dt u ⎪ ⎪ ⎪ 0 ⎪ ⎨ x 11 2x 3 7. + e (x) = − + ((x − t − 1)u(t) − (x − t + 1)v(t)) dt v ⎪ ⎪ ⎪ 2 2 0 ⎪ ⎪ ⎪ ⎩ u(0) = 1, v(0) = 2 ⎧ x ⎪ x ⎪ (x) = −3x + e + (u(t) + v(t)) dt, u(0) = 2 u ⎪ ⎨ 8.

0

⎪ ⎪ ⎪ ⎩ v (x) = 2 + x2 − 3ex +

x 0

(u(t) − (x − t)v(t)) dt, v(0) = 1

x ⎧ 2 4 ⎪ x ⎪ x u (x) = 3 + − e + ((1 − xt)u(t) − (1 + xt)v(t)) dt ⎪ ⎪ ⎪ 3 0 ⎨ x 9. 2 ⎪ ((1 + xt)u(t) − (1 − xt)v(t)) dt v (x) = 3 − x4 − 3ex + ⎪ ⎪ ⎪ 3 0 ⎪ ⎩ u(0) = 1, v(0) = −1 x ⎧ 1 2 ⎪ x ⎪ (x) = −x − + e + (v(t) + w(t)) dt, u(0) = 2 x u ⎪ ⎪ ⎪ 2 0 ⎪ ⎪ ⎪ x ⎨ 1 v (x) = x − x2 − ex + (w(t) − u(t)) dt, v(0) = 0 10. ⎪ 2 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (u(t) − v(t)) dt, w(0) = 1 ⎩ w (x) = 3 − ex + 0

10.4 Systems of Volterra Integro-Diﬀerential Equations ⎧ x 1 2 ⎪ ⎪ (x) = −x − − sin x + (v(t) + w(t)) dt, u(0) = 2 x u ⎪ ⎪ ⎪ 2 0 ⎪ ⎪ ⎪ x ⎨ 1 (w(t) − u(t)) dt, v(0) = 0 v (x) = x − x2 + sin x + 11. ⎪ 2 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (u(t) − v(t)) dt, w(0) = 1 ⎩ w (x) = 1 − 3 sin x +

335

0

⎧ x ⎪ ⎪ (u(t) − v(t) + w(t)) dt, u(0) = 2 u (x) = cos x − 2 sin x − ex + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ x ⎨ (u(t) − v(t) − w(t)) dt, v(0) = 1 v (x) = −2 − sin x + ex + 12. ⎪ 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (v(t) + w(t) − u(t)) dt, w(0) = 1 ⎩ w (x) = 2 + sin x − cos x + 0

10.4.2 The Laplace Transform Method The Laplace transform method was presented in Chapter 1, and was used in other chapters and in this chapter as well. Before we start applying this method, we summarize some of the concepts presented in this text. The Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x L{(f1 ∗ f2 )(x)} = L f1 (x − t)f2 (t)dt = F1 (s)F2 (s). (10.99) 0

Moreover, the Laplace Transforms of Derivatives can be summarized as follows: L{f (x)} = sL{f (x)} − f (0), L{f (x)} = s2 L{f (x)} − sf (0) − f (0), (10.100) 3 2 L{f (x)} = s L{f (x)} − s f (0) − sf (0) − f (0), and so on. Based on this summary, we will examine speciﬁc Volterra integrodiﬀerential equations where the kernel is a diﬀerence kernel. Recall that we will apply the Laplace transform method and the inverse of the Laplace transform using Table 1.1 in section 1.5. The Laplace transform method for solving systems of Volterra integro-diﬀerential equations will be illustrated by studying the following examples. Example 10.17 Solve the system of Volterra integro-diﬀerential equations by using the Laplace transform method x ((x − t)u(t) + (x − t)v(t)) dt, u(0) = 1 u (x) = 2x2 + 0 (10.101) x 1 v (x) = −3x2 − x5 + ((x − t)u(t) − (x − t)v(t)) dt, v(0) = 1. 10 0

336

10 Systems of Volterra Integral Equations

Notice that the kernels K1 (x − t) = K2 (x − t) = x − t. Taking Laplace transform of both sides of each equation gives 4 1 1 sU (s) − 1 = 3 + 2 U (s) + 2 V (s), s s s (10.102) 12 1 1 6 sV (s) − 1 = − 3 − 6 + 2 U (s) − 2 V (s), s s s s or equivalently 1 1 4 s − 2 U (s) − 2 V (s) = 1 + 3 , s s s (10.103) 6 12 1 1 s + 2 V (s) − 2 U (s) = 1 − 3 − 6 . s s s s Solving this system of equations for U (s) and V (s) gives 1 1 3! 3! (10.104) U (s) = + 4 , V (s) = − 4 . s s s s By taking the inverse Laplace transform of both sides of each equation, the exact solutions are given by (u(x), v(x)) = (1 + x3 , 1 − x3 ). (10.105) Example 10.18 Solve the system of Volterra integro-diﬀerential equations by using the Laplace transform method x (cos(x − t)u(t) + cos(x − t)v(t)) dt, u(0) = 1 u (x) = 1 − 2 sin x + 0

v (x) = −3 + 2 cos x +

0

x

(sin(x − t)u(t) + sin(x − t)v(t)) dt, v(0) = 1.

(10.106) Taking Laplace transform of both sides of each equation gives 1 2 s s sU (s) − 1 = − 2 + U (s) + 2 V (s), s s + 1 s2 + 1 s +1 (10.107) 3 2s 1 1 sV (s) − 1 = − + 2 + 2 U (s) + 2 V (s). s s +1 s +1 s +1 Solving this system of equations for U (s) and V (s) gives 1 1 1 1 (10.108) U (s) = + 2 , V (s) = − 2 . s s s s By taking the inverse Laplace transform of both sides of each equation, the exact solutions are given by (u(x), v(x)) = (1 + x, 1 − x). (10.109) Example 10.19 Solve the system of Volterra integro-diﬀerential equations by using the Laplace transform method

10.4 Systems of Volterra Integro-Diﬀerential Equations

u (x) = cos x − 2 sin x + v (x) = cos x + x cos x +

337

x

(cos(x − t)u(t) + sin(x − t)v(t)) dt,

0

x

(sin(x − t)u(t) + sin(x − t)v(t)) dt,

0

(10.110)

u(0) = 1, v(0) = −1. Taking Laplace transform of both sides of each equation gives s 2 s 1 sU (s) − 1 = 2 − + U (s) + 2 V (s), s + 1 s2 + 1 s2 + 1 s +1 (10.111) 1 s2 − 1 1 s + + 2 U (s) + V (s). sV (s) + 1 = 2 s + 1 (s + 1)2 s2 + 1 s2 + 1 Solving this system of equations for U (s) and V (s) gives s 1 s 1 (10.112) + , V (s) = 2 − . U (s) = 2 s + 1 s2 + 1 s + 1 s2 + 1 By taking the inverse Laplace transform of both sides of each equation, the exact solutions are given by (u(x), v(x)) = (sin x + cos x, sin x − cos x).

(10.113)

Example 10.20 Solve the system of Volterra integro-diﬀerential equations by using the Laplace transform method x u (x) = ex − e2x + e4x + (2v(t) − 4w(t)) dt, u(0) = 1, u (0) = 1, 0

v (x) = ex + 4e2x − e4x +

2x

w (x) = −e + e x

+ 16e

4x

x

0

(4w(t) − u(t)) dt, v(0) = 1, v (0) = 2,

+ 0

x

(u(t) − 2v(t)) dt, w(0) = 1, w (0) = 4.

(10.114) Taking Laplace transform of both sides of each equation gives 1 1 1 2 4 s2 U (s) − s − 1 = − + + V (s) − W (s), s−1 s−2 s−4 s s 1 4 1 4 1 (10.115) + − + W (s) − U (s), s2 V (s) − s − 2 = s−1 s−2 s−4 s s 1 1 16 1 2 s2 W (s) − s − 4 = − + + + U (s) − V (s). s−1 s−2 s−4 s s Solving this system of equations for U (s) and V (s) gives 1 1 1 (10.116) , V (s) = , W (s) = . U (s) = s−1 s−2 s−4 By taking the inverse Laplace transform of both sides of each equation, the exact solutions are given by (u(x), v(x), w(x)) = (ex , e2x , e4x ). (10.117)

338

10 Systems of Volterra Integral Equations

Exercises 10.4.2 Use the Laplace transform method to solve the following systems of Volterra integrodiﬀerential equations ⎧ x 1 3 ⎪ (x) = −x2 + ⎪ x u + ((x − t)u(t) + (x − t + 1)v(t)) dt, u(0) = 1 ⎪ ⎨ 6 0 1. x ⎪ 1 4 ⎪ ⎪ ((x − t)u(t) − (x − t)v(t)) dt, v(0) = 1 x + ⎩ v (x) = −x − 12 0 ⎧ x 1 ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt, u(0) = 0 ⎪ u (x) = 1 + x − x3 + ⎨ 3 0 2. x ⎪ 1 4 ⎪ ⎪ x + ((x − t)u(t) − (x − t)v(t)) dt, v(0) = 0 ⎩ v (x) = 1 − x − 12 0 x ⎧ ⎪ ⎪ (cos(x − t)u(t) + cos(x − t)v(t)) dt u (x) = 2 − 2 sin x + ⎪ ⎪ ⎪ 0 ⎨ x 3. ⎪ (x) = −4 + 2 cos x + (sin(x − t)u(t) + sin(x − t)v(t)) dt v ⎪ ⎪ ⎪ 0 ⎪ ⎩ u(0) = 1, v(0) = 1 ⎧ x

x−t ⎪ x ⎪ (x) = 3 + 3x − 2e + u(t) + ex−t v(t) dt, u(0) = 0 u e ⎪ ⎨ 4.

0

⎪ ⎪ ⎪ ⎩ v (x) = 3 + x + x2 − 2ex + ⎧ ⎪ ⎪ ⎪ u (x) = 2 cos x − ex + ⎨

5.

x

0

⎪ ⎪ ⎪ ⎩ v (x) = 1 − x − cos x +

x

0

x 0

x−t u(t) − ex−t v(t) dt, v(0) = 0 e

ex−t u(t) + ex−t v(t) dt, u(0) = 0 ((x − t)u(t) − (x − t)v(t)) dt, v(0) = 1

x ⎧ ⎪ ⎪ (cos(x − t)u(t) + sin(x − t)v(t)) dt u (x) = cos x − x sin x + ⎪ ⎪ ⎪ 0 ⎨ x 6. ⎪ (x) = −2 sin x + (sin(x − t)u(t) + cos(x − t)v(t)) dt v ⎪ ⎪ ⎪ 0 ⎪ ⎩ u(0) = 0, v(0) = 1 ⎧ x 3 2 ⎪ ⎪ (x) = −2x − + cos x + (u(t) + (x − t)v(t)) dt, u(0) = 2 u x ⎪ ⎨ 2 0 7. x ⎪ ⎪ ⎪ ((x − t)u(t) + v(t)) dt, v(0) = 2 ⎩ v (x) = −4x − x2 + 3 sin x + ⎧ ⎪ x ⎪ u (x) = 5 − 4e + ⎪ ⎨ 8.

0

x

0

⎪ ⎪ ⎪ ⎩ v (x) = −1 − 2xex +

ex−t u(t) + ex−t v(t) dt, u(0) = 3 x

0

⎧ ⎪ ⎪ ⎪ u (x) = 3 + 2x − 3ex + ⎨ 9.

x−t

e u(t) − ex−t v(t) dt, v(0) = 2

x−t u(t) + ex−t v(t) dt, u(0) = −1 e

x

0

⎪ ⎪ ⎪ ⎩ v (x) = 1 + ex + 2xex +

x

0

x−t u(t) − ex−t v(t) dt, v(0) = 1 e

References ⎧ x 1 2 ⎪ ⎪ (x) = −x − − sin x + (v(t) + w(t)) dt, u(0) = 2 x u ⎪ ⎪ ⎪ 2 0 ⎪ ⎪ ⎪ x ⎨ 1 (w(t) − u(t)) dt, v(0) = 0 v (x) = x − x2 + sin x + 10. ⎪ 2 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (u(t) − v(t)) dt, w(0) = 1 ⎩ w (x) = 1 − 3 sin x +

339

0

⎧ x ⎪ ⎪ (u(t) − v(t) + w(t)) dt, u(0) = 2 u (x) = cos x − 2 sin x − ex + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ x ⎨ (u(t) − v(t) − w(t)) dt, v(0) = 1 v (x) = −2 − sin x + ex + 11. ⎪ 0 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ (v(t) + w(t) − u(t)) dt, w(0) = 1 ⎩ w (x) = 2 + sin x − cos x + 0

⎧ x ⎪ (x) = −x3 − x4 + ⎪ (3v(t) + 4w(t)) dt, u(0) = 0, u (0) = 1 u ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ x ⎨ (4w(t) − 2u(t)) dt, v(0) = 0, v (0) = 0 v (x) = 2 + x2 − x4 + 12. ⎪ 0 ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ (2u(t) − 3v(t)) dt, w(0) = 0, w (0) = 0 ⎩ w (x) = 6x − x2 + x3 + 0

References 1. A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 2. P. Linz, A simple approximation method for solving Volterra integro-diﬀerential equations of the ﬁrst kind, J. Inst. Maths. Appl., 14 (1974) 211–215. 3. P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, (1985). 4. D. Porter and D.S. Stirling, Integral Equations: a Practical Treatment from Spectral Theory to Applications, Cambridge, (2004). 5. G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994). 6. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 7. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 8. J.H. He, Variational iteration method for autonomous ordinary diﬀerential systems, Appl. Math. Comput., 114(2/3) (2000) 115–123.

Chapter 11

Systems of Fredholm Integral Equations

11.1 Introduction Systems of Volterra and Fredholm integral equations have attracted much concern in applied sciences. The systems of Fredholm integral equations appear in two kinds. The system of Fredholm integral equations of the ﬁrst kind [1–5] reads b ˜ 1 (x, t)v(t) dt, f1 (x) = K1 (x, t)u(t) + K a (11.1) b ˜ K2 (x, t)u(t) + K2 (x, t)v(t) dt, f2 (x) = a

where the unknown functions u(x) and v(x) appear only under the integral sign, and a and b are constants. However, for systems of Fredholm integral equations of the second kind, the unknown functions u(x) and v(x) appear inside and outside the integral sign. The second kind is represented by the form b ˜ 1 (x, t)v(t) dt, u(x) = f1 (x) + K1 (x, t)u(t) + K a

v(x) = f2 (x) +

b

˜ 2 (x, t)v(t) dt. K2 (x, t)u(t) + K

(11.2)

a

The systems of Fredholm integro-diﬀerential equations have also attracted a considerable size of interest. These systems are given by b ˜ 1 (x, t)v(t) dt, K1 (x, t)u(t) + K u(i) (x) = f1 (x) + a (11.3) b (i) ˜ K2 (x, t)u(t) + K2 (x, t)v(t) dt, v (x) = f2 (x) + a

where the initial conditions for the last system should be prescribed.

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

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11 Systems of Fredholm Integral Equations

11.2 Systems of Fredholm Integral Equations In this section we will study systems of Fredholm integral equations of the second kind given by b ˜ 1 (x, t)v(t) + · · · dt, u(x) = f1 (x) + K1 (x, t)u(t) + K a

v(x) = f2 (x) + .. .

b

˜ 2 (x, t)v(t) + · · · dt, K2 (x, t)u(t) + K

(11.4)

a

The unknown functions u(x), v(x), . . . that will be determined, appear inside ˜ i (x, t), and the funcand outside the integral sign. The kernels Ki (x, t) and K tion fi (x) are given real-valued functions. In what follows we will present the methods, new and traditional, that will be used. Recall that the Fredholm integral equations were presented in Chapter 4 where a variety of methods, new and traditional, were applied. In this section, we will focus our study only on two methods, namely the Adomian decomposition method and the direct computation method.

11.2.1 The Adomian Decomposition Method The Adomian decomposition method [6–7], as presented before, decomposes each solution as an inﬁnite sum of components, where these components are determined recurrently. This method can be used in its standard form, or combined with the noise terms phenomenon. Moreover, the modiﬁed decomposition method will be used wherever it is appropriate. Example 11.1 Use the Adomian decomposition method to solve the following system of Fredholm integral equations π u(x) = sin x − 2 − 2x − πx + ((1 + xt)u(t) + (1 − xt)v(t)) dt, 0 (11.5) π ((1 − xt)u(t) − (1 + xt)v(t)) dt. v(x) = cos x − 2 − 2x + πx + 0

The Adomian decomposition method suggests that the linear terms u(x) and v(x) be decomposed by an inﬁnite series of components ∞ ∞ un (x), v(x) = vn (x), u(x) = (11.6) n=0

n=0

where un (x) and vn (x), n 0 are the components of u(x) and v(x) that will be elegantly determined in a recursive manner.

11.2 Systems of Fredholm Integral Equations

343

Substituting (11.6) into (11.5) gives ∞ un (x) = sin x − 2 − 2x − πx n=0 π ∞ ∞ (1 + xt) un (t) + (1 − xt) vn (t) dt, + 0

∞ n=0

n=0

n=0

(11.7)

vn (x) = cos x − 2 − 2x + πx π ∞ ∞ un (t) − (1 + xt) vn (t) dt. + (1 − xt) 0

n=0

n=0

The modiﬁed decomposition method will be used here, hence we set the recursive relation u0 (x) = sin x − 2, v0 (x) = cos x − 2, π ((1 + xt)u0 (t) + (1 − xt)v0 (t)) dt, u1 (x) = −2x − πx + 0

v1 (x) = −2x + πx + uk+1 (x) = vk+1 (x) =

0

π

((1 − xt)u0 (t) − (1 + xt)v0 (t)) dt,

(11.8)

π 0

((1 − xt)uk (t) − (1 + xt)vk (t)) dt,

k 1,

((1 − xt)uk (t) − (1 + xt)vk (t)) dt,

k 1.

x 0

This in turn gives u0 (x) = sin x − 2, and v0 (x) = cos x − 2,

u1 (x) = 2 − 4π,

(11.9)

v1 (x) = 2 + 2π 2 x.

(11.10)

By canceling the noise terms ∓2 from u0 (x) and from v0 (x) we obtain the exact solutions (u(x), v(x)) = (sin x, cos x). (11.11) Example 11.2 Use the Adomian decomposition method to solve the following system of Fredholm integral equations π4 π3 2 + (tu(t) + tv(t)) dt, u(x) = x + sec x − 96 0 (11.12) π 4 π2 2 + (u(t) + v(t)) dt. v(x) = x − sec x − 16 0 Proceeding as before we obtain

344

11 Systems of Fredholm Integral Equations

∞

π3 + un (x) = x + sec2 x − 96 n=0 ∞

π2 + vn (x) = x − sec2 x − 16 n=0

π 4

0

0

π 4

t

∞

un (t) + t

n=0 ∞

un (t) +

n=0

∞

vn (t) dt,

n=0 ∞

(11.13)

vn (t) dt.

n=0

The modiﬁed decomposition method will be used here, hence we set the recursive relation u0 (x) = x + sec2 x, v0 (x) = x − sec2 x, π4 π3 u1 (x) = − (tu0 (t) + tv0 (t)) dt, + 96 0 π4 π2 (u0 (t) + v0 (t)) dt, v1 (x) = − + 16 (11.14) 0 π4 uk+1 (x) = (tuk (t) + tvk (t)) dt, k 1, vk+1 (x) =

0

π 4

0

(uk (t) + vk (t)) dt,

k 1.

This in turn gives and

u0 (x) = x + sec2 x,

u1 (x) = 0,

(11.15)

v0 (x) = x − sec2 x,

v1 (x) = 0.

(11.16)

As a result, the remaining components uk (x) and vk (x) for k 2 vanish. The exact solutions are given by (11.17) (u(x), v(x)) = (x + sec2 x, x − sec2 x). Example 11.3 Use the Adomian decomposition method to solve the following system of Fredholm integral equations 1 xt

2 x sinh(1 + x) + e u(t) + e−xt v(t) dt, u(x) = e − 1+x 0 (11.18) 1 −xt

2 xt −x sinh(1 − x) + e u(t) + e v(t) dt. v(x) = e − 1−x 0 Proceeding as before and using the modiﬁed decomposition method, we ﬁnd the recursive relation

11.2 Systems of Fredholm Integral Equations

v0 (x) = e−x ,

u0 (x) = ex ,

2 u1 (x) = sinh(1 + x) + 1+x v1 (x) = − uk+1 (x) = vk+1 (x) =

2 sinh(1 − x) + 1−x 1

1

ext u0 (t) + e−xt v0 (t) dt = 0,

0

0

1

e−xt u0 (t) + ext v0 (t) dt = 0,

ext uk (t) + e−xt vk (t) dt = 0,

0

0

1

345

e−xt uk (t) + ext vk (t) dt = 0,

(11.19)

k 1, k 1.

The exact solutions are given by (u(x), v(x)) = (ex , e−x ).

(11.20)

Example 11.4 Use the Adomian decomposition method to solve the following system of Fredholm integral equations 1 4 u(x) = x − + (v(t) + w(t)) dt, 3 −1 1 2 2 (11.21) (w(t) + u(t)) dt, v(x) = x + x − + 3 −1 1 2 w(x) = x2 + x3 − + (u(t) + v(t)) dt. 3 −1 Proceeding as before and using the modiﬁed decomposition method, we ﬁnd the recursive relation u0 (x) = x, v0 (x) = x + x2 , w0 (x) = x2 + x3 , 1 4 (v0 (t) + w0 (t)) dt = 0, u1 (x) = − + 3 −1 (11.22) 1 2 v1 (x) = − + (w0 (t) + u0 (t)) dt = 0, 3 −1 1 2 (u0 (t) + v0 (t)) dt = 0, w1 (x) = − + 3 −1 uk+1 (x) = 0, vk+1 = 0, wk+1 = 0, k 1. The exact solutions are given by (u(x), v(x), w(x)) = (x, x + x2 , x2 + x3 ).

(11.23)

346

11 Systems of Fredholm Integral Equations

Exercises 11.2.1 Solve the following systems of Fredholm integral equations 1 ⎧ 16 ⎪ ⎪ (tu(t) + tv(t)) dt + u(x) = x − ⎪ ⎨ 15 −1 1. 1 ⎪ 2 ⎪ ⎪ (xu(t) + xv(t)) dt ⎩ v(x) = x2 + x3 − x + 3 −1 1 ⎧ 2

2 2 2 ⎪ ⎪ x + (x − t2 )u(t) + (x2 − t2 )v(t) dt + u(x) = x − ⎪ ⎨ 3 5 −1 2. 1 ⎪

2 2 ⎪ ⎪ (x − t2 )u(t) − xtv(t) dt ⎩ v(x) = x2 + x3 + x + 5 −1 1 ⎧ 16 ⎪ 2 ⎪ − (u(t) + v(t)) dt u(x) = x + x + ⎪ ⎨ 15 −1 3. 1 ⎪ 16 2 ⎪ ⎪ ((x − t)u(t) + xtv(t)) dt x+ + ⎩ v(x) = x3 + x4 − 15 3 −1 ⎧ π ⎪ ⎪ (u(t) + v(t)) dt ⎪ ⎨ u(x) = sin x + cos x − 4 + 0 4. π ⎪ ⎪ ⎪ ((x − t)u(t) + (x − t)v(t)) dt ⎩ v(x) = sin x − cos x − 4x + 2π + 0

π ⎧ ⎪ 2 2 ⎪ x − π + (u(t) + v(t)) dt u(x) = x + sin ⎪ ⎨ 0 5. π ⎪ π2 ⎪ ⎪ ⎩ v(x) = x − cos2 x − π 2 x + ((x − t)u(t) + (x + t)v(t)) dt + 2 0 π ⎧ ⎪ 2 ⎪ ((x − t)u(t) − (x + t)v(t)) dt ⎪ ⎨ u(x) = x + sin x + 2π + 6.

−π

⎪ ⎪ ⎪ ⎩ v(x) = x2 + cos x − 2π + ⎧ ⎪ ⎪ ⎪ ⎨ u(x) =

2−π 2

π

−π

((x + t)u(t) − (x − t)v(t)) dt

x + x tan−1 x +

1

−1

(xu(t) + xv(t)) dt

1 ⎪ 3π − 2 ⎪ ⎪ + x + tan−1 x + ⎩ v(x) = (tu(t) − tv(t)) dt 6 −1 ⎧ 1 ex ⎪ ⎪ u(x) = −x + + (xu(t) + xv(t)) dt ⎪ ⎨ 1 + ex 0 8. 1 ⎪ 1 ⎪ ⎪ ⎩ v(x) = −1 + + (u(t) + v(t)) dt 1 + ex 0 ⎧ π sin x 2 ⎪ ⎪ + u(x) = (2 − π)x + (xu(t) + xv(t)) dt ⎪ ⎨ 1 + sin x 0 9. π ⎪ cos x ⎪ 2 ⎪ ⎩ v(x) = (2 − π) + (u(t) + v(t)) dt + 1 + cos x 0

7.

11.2 Systems of Fredholm Integral Equations

347

√ ⎧ π 3− 3 6 ⎪ ⎪ + sec x tan x + u(x) = (u(t) − v(t)) dt ⎪ ⎨ 3 0 10. π ⎪ √ ⎪ 6 ⎪ ⎩ v(x) = (1 − 3)x + sec2 x + (xu(t) + xv(t)) dt 0

1 ⎧ 1 ⎪ ⎪ (v(t) − w(t)) dt + u(x) = x − ⎪ ⎪ 12 ⎪ −1 ⎪ ⎪ ⎪ 1 ⎨ 1 v(x) = x2 + + (w(t) − u(t)) dt 11. ⎪ 4 −1 ⎪ ⎪ ⎪ 1 ⎪ ⎪ 1 ⎪ ⎪ ⎩ w(x) = x2 − + (u(t) − v(t)) dt 6 −1 π ⎧ π 5π 4 ⎪ 2 ⎪ + sec x + (v(t) − w(t)) dt u(x) = 1 + ⎪ ⎪ 8 8 ⎪ 0 ⎪ ⎪ ⎪ π ⎨ π 3π 4 12. − sec2 x + v(x) = 1 − (w(t) − u(t)) dt ⎪ 8 8 ⎪ 0 ⎪ ⎪ ⎪ π ⎪ ⎪ π π ⎪ ⎩ w(x) = 1 − + sec2 x + 4 (u(t) − v(t)) dt 4 2 0

11.2.2 The Direct Computation Method It was stated before that all methods that we used in Chapter 4 to handle Fredholm integral equations can be used here to solve systems of Fredholm integral equations. In the previous section we selected the Adomian decomposition method. In this section we will employ the direct computational method. The other methods such as the variational iteration method, successive substitution method, and others can be used as well. The direct computational method will be applied to solve the systems of Fredholm integral equations of the second kind. The method was used before in this text, therefore we summarize the necessary steps. The method will be applied for the degenerate or separable kernels of the form n n gk (x)hk (t), K2 (x, t) = rk (x)sk (t). (11.24) K1 (x, t) = k=1

k=1

The direct computation method can be applied as follows: 1. We ﬁrst substitute (11.24) into the system of Fredholm integral equations the form b ˜ 1 (x, t)v(t) dt, K1 (x, t)u(t) + K u(x) = f1 (x) + a (11.25) b ˜ K2 (x, t)u(t) + K2 (x, t)v(t) dt. v(x) = f2 (x) + a

2. This substitution gives

348

11 Systems of Fredholm Integral Equations

u(x) = f1 (x) + v(x) = f2 (x) +

n k=1 n k=1

b

gk (x)

hk (t)u(t)dt +

a b

rk (x)

sk (t)u(t)dt + a

n k=1 n

b

g˜k (x)

(11.26)

b

r˜k (x)

k=1

˜ k (t)v(t)dt, h

a

s˜k (t)v(t)dt. a

3. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (11.26) becomes u(x) = f1 (x) + α1 g1 (x) + · · · + αn gn (x) + β1 g˜1 (x) + · · · + βn g˜n (x), (11.27) v(x) = f2 (x) + γ1 r1 (x) + · · · + γn rn (x) + δ1 r˜1 (x) + · · · + δn r˜n (x), where b

hi (t)u(t)dt, 1 i n,

αi = a

b

βi =

˜ i (t)v(t)dt, 1 i n, h

a

(11.28)

b

si (t)u(t)dt, 1 i n,

γi = a

b

s˜i (t)v(t)dt, 1 i n.

δi = a

4. Substituting (11.27) into (11.28) gives a system of n algebraic equations that can be solved to determine the constants αi , βi , γi , and δi . To facilitate the computational work, we can use the computer symbolic systems such as Maple and Mathematica. Using the obtained numerical values of these constants into (11.27), the solutions u(x) and v(x) of the system of Fredholm integral equations (11.25) follow immediately. Example 11.5 Solve the following system of Fredholm integral equations by using the direct computation method π u(x) = sin x + cos x − 4x + (xu(t) + xv(t))dt, 0 (11.29) π v(x) = sin x − cos x + (u(t) − v(t))dt. 0

Following the analysis presented above this system can be rewritten as u(x) = sin x + cos x + (α − 4)x, (11.30) v(x) = sin x − cos x + β, where π

π

(u(t) + v(t))dt,

α= 0

β= 0

(u(t) − v(t))dt.

To determine α, and β, we substitute (11.30) into (11.31) to obtain

(11.31)

11.2 Systems of Fredholm Integral Equations

349

π2 π2 α + πβ, β = −2π 2 + α − πβ. 2 2 Solving this system gives α = 4, β = 0. α = (4 − 2π2 ) +

Substituting (11.33) into (11.30) leads to the exact solutions (u(x), v(x)) = (sin x + cos x, sin x − cos x).

(11.32) (11.33) (11.34)

Example 11.6 Solve the following system of Fredholm integral equations by using the direct computation method π3 (tan tu(t) + sec tv(t))dt, u(x) = sec x − 2 + 0 (11.35) π3 √ (sec tu(t) + sec tv(t))dt. v(x) = −(1 + 3) + tan x + 0

Following the analysis presented above this system can be rewritten as u(x) = sec x − 2 + α1 + β1 , √ (11.36) v(x) = tan x − (1 + 3) + α2 + β1 , where π3 π3 π3 α1 = tan tu(t)dt, α2 = sec tu(t)dt, β1 = sec tv(t)dt. (11.37) 0

0

0

To determine α1 , α2 , and β1 , we substitute (11.36) into (11.37) and solve the resulting system we ﬁnd √ α1 = 1, α2 = 3, β1 = 1. (11.38) Substituting (11.38) into (11.36) leads to the exact solutions (u(x), v(x)) = (sec x, tan x).

(11.39)

Example 11.7 Solve the following system of Fredholm integral equations by using the direct computation method 1 (ln(xt)u(t) + ln(xt2 )v(t))dt, u(x) = 6 − ln x + 0+ (11.40) 1 v(x) = −4 + ln x +

0+

(ln(xt2 )u(t) − ln(xt)v(t))dt.

Proceeding as before, this system can be rewritten as u(x) = 6 + (α1 + β1 − 1) ln x + α2 + 2β2 , v(x) = −4 + (α1 − β1 + 1) ln x + 2α2 − β2 , where

(11.41)

350

11 Systems of Fredholm Integral Equations

α1 = β1 =

1

u(t)dt, 0+ 1

v(t)dt, 0+

α2 = β2 =

1

ln(t)u(t)dt, 0+ 1

(11.42) ln(t)v(t)dt.

0+

To determine αi , and βi , 1 i 2, we substitute (11.41) into (11.42) to obtain α2 = −8 + 2α1 − α2 + 2β1 − 2β2 , α1 = 7 − α1 + α2 − β1 + 2β2 , (11.43) β1 = −5 − α1 + 2α2 + β1 − β2 , β2 = 6 + 2α1 − 2α2 − 2β1 + β2 . Solving this system gives α1 = 0, α2 = 1, β1 = 2, β2 = −3. (11.44) These results lead to the exact solutions (u(x), v(x)) = (1 + ln x, 1 − ln x).

(11.45)

Example 11.8 Solve the following system of Fredholm integral equations by using the direct computation method π4 2 (v(t) − w(t))dt, u(x) = + sec2 x + 3 0 π4 10 (11.46) (w(t) − u(t))dt, v(x) = − sec2 x + 3 0 π4 4 w(x) = −1 − sec x + (u(t) − v(t))dt. 0

Proceeding as before, this system can be rewritten as 2 u(x) = + sec2 x + β − γ, 3 10 − sec2 x + γ − α, v(x) = 3 w(x) = −1 − sec4 x + α − β, where π4 π4 π4 u(t)dt, β = v(t)dt, γ = w(t)dt. α= 0

0

(11.47)

(11.48)

0

To determine α, β, and γ, we substitute (11.47) into (11.48) and by solving the resulting system we ﬁnd π π π 4 α = + 1, β = − 1, γ = − . (11.49) 4 4 4 3 These results lead to the exact solutions (11.50) (u(x), v(x), w(x)) = (1 + sec2 x, 1 − sec2 x, 1 − sec4 x).

11.2 Systems of Fredholm Integral Equations

351

Exercises 11.2.2 Use the direct computation method to solve the following systems of Fredholm integral equations 1 ⎧ 16 ⎪ ⎪ u(x) = x − (tu(t) + tv(t)) dt + ⎪ ⎨ 15 −1 1. 1 ⎪ 2 ⎪ ⎪ (xu(t) + xv(t)) dt ⎩ v(x) = x2 + x3 − x + 3 −1 ⎧ 1

⎪ ⎪ ln tu(t) − ln(xt2 )v(t) dt ⎪ ⎨ u(x) = −6 + 4 ln x + 2.

0+

⎪ ⎪ ⎪ ⎩ v(x) = −2 − 2 ln x +

1

0+

ln(xt2 )u(t) − ln tv(t) dt

1 ⎧ 16 ⎪ 2 ⎪ ⎪ ⎨ u(x) = x + x − 15 + −1 (u(t) + v(t)) dt 3. 1 ⎪ 2 16 ⎪ ⎪ x+ + ((x − t)u(t) + xtv(t)) dt ⎩ v(x) = x3 + x4 − 15 3 −1 ⎧ π ⎪ ⎪ (tu(t) + tv(t)) dt u(x) = sin x + cos x − 2π + ⎪ ⎨ 4.

0

⎪ ⎪ ⎪ ⎩ v(x) = sin x − cos x + πx +

π 0

(xu(t) − xtv(t)) dt

π ⎧ ⎪ 2 2 ⎪ x − π + (u(t) + v(t)) dt u(x) = x + sin ⎪ ⎨ 0 5. π ⎪ π2 ⎪ ⎪ ((x − t)u(t) + (x + t)v(t)) dt + ⎩ v(x) = x − cos2 x − π 2 x + 2 0 ⎧ π 3 ⎪ ⎪ u(x) = −2 + tan x + (sec tu(t) + tan tv(t)) dt ⎪ ⎨ 0

6.

π ⎪ π ⎪ 3 ⎪ ⎩ v(x) = − + sec x + (tan tu(t) − sec tv(t)) dt 3 0 π ⎧ ⎪ ⎪ u(x) = x2 + sin x + 2π + ((x − t)u(t) − (x + t)v(t)) dt ⎪ ⎨

7.

⎪ ⎪ ⎪ ⎩ v(x) = x2 + cos x − 2π +

−π

π

−π

((x + t)u(t) − (x − t)v(t)) dt

⎧ 1 ex ⎪ ⎪ + (xu(t) + xv(t)) dt u(x) = −x + ⎪ ⎨ 1 + ex 0 8. 1 ⎪ 1 ⎪ ⎪ ⎩ v(x) = −1 + + (u(t) + v(t)) dt 1 + ex 0 ⎧ π sin x 2 ⎪ ⎪ (xu(t) + xv(t)) dt u(x) = (2 − π)x + + ⎪ ⎨ 1 + sin x 0 9. π ⎪ cos x ⎪ 2 ⎪ ⎩ v(x) = (2 − π) + (u(t) + v(t)) dt + 1 + cos x 0

352

11 Systems of Fredholm Integral Equations

√ ⎧ π 3− 3 6 ⎪ ⎪ + sec x tan x + u(x) = (u(t) − v(t)) dt ⎪ ⎨ 3 0 10. π ⎪ √ ⎪ 6 ⎪ ⎩ v(x) = (1 − 3)x + sec2 x + (xu(t) + xv(t)) dt 0

1 ⎧ 1 ⎪ ⎪ (v(t) − w(t)) dt + u(x) = x − ⎪ ⎪ 12 ⎪ −1 ⎪ ⎪ ⎪ 1 ⎨ 1 v(x) = x2 + + (w(t) − u(t)) dt 11. ⎪ 4 −1 ⎪ ⎪ ⎪ 1 ⎪ ⎪ 1 ⎪ ⎪ ⎩ w(x) = x2 − + (u(t) − v(t)) dt 6 −1 π ⎧ √ π 4 ⎪ 2 ⎪ + x + 2 − (v(t) − w(t)) dt u(x) = 1 + sec ⎪ ⎪ 4 ⎪ 0 ⎪ ⎪ ⎪ π ⎨ √ π 4 12. v(x) = 1 − sec2 x − 2 − + (w(t) + u(t)) dt ⎪ 4 ⎪ 0 ⎪ ⎪ ⎪ π ⎪ ⎪ π ⎪ ⎩ w(x) = sec x tan x − + 4 (u(t) + v(t)) dt 2 0

11.3 Systems of Fredholm Integro-Diﬀerential Equations In Chapter 6, the Fredholm integro-diﬀerential equations were studied. The Fredholm integro-diﬀerential equations, were given in the form x K(x, t)u(t)dt, (11.51) u(i) (x) = f (x) + λ 0 i

where u(i) (x) = ddxui . Because the resulted equation combines the diﬀerential operator and the integral operator, then it is necessary to deﬁne initial conditions u(0), u (0), . . . , u(i−1) (0) for the determination of the particular solution u(x) of the Fredholm integro-diﬀerential equation (11.51). In this section, we will study systems of Fredholm integro-diﬀerential equations of the second kind given by b (i) ˜ 1 (x, t)v(t) dt, K1 (x, t)u(t) + K u (x) = f1 (x) + a (11.52) b ˜ 2 (x, t)v(t) dt. K2 (x, t)u(t) + K v (i) (x) = f2 (x) + a

The unknown functions u(x), v(x), . . ., that will be determined, occur inside the integral sign whereas the derivatives of u(x), v(x), . . . appear mostly out˜ i (x, t), and the function side the integral sign. The kernels Ki (x, t) and K fi (x) are given real-valued functions. In Chapter 6, four analytical methods were used for solving the Fredholm integro-diﬀerential equations. These methods are the variational iteration method (VIM), the Adomian decomposition method (ADM), the direct

11.3 Systems of Fredholm Integro-Diﬀerential Equations

353

computation method, and the series solution method. The aforementioned methods can eﬀectively handle the systems of Fredholm integro-diﬀerential equations (11.52). However, in this section, we will use only two of these methods, namely the direct computation method and the variational iteration method. The reader can apply the other two methods to show that it can be used to handle the system (11.52).

11.3.1 The Direct Computation Method The direct computation method is a reliable technique that was used before to handle Fredholm integral equations in Chapter 4, Fredholm integrodiﬀerential equations in Chapter 6, and systems of Fredholm integral equations in this chapter. The direct computation method will be applied to solve the systems of Fredholm integro-diﬀerential equations of the second kind in a parallel manner to our treatment that was used before. The method approaches any Fredholm equation in a direct manner and gives the solution in an exact form and not in a series form. The method will be applied for the degenerate or separable kernels of the form n n ˜ k (t), ˜ 1 (x, t) = gk (x)hk (t), K g˜k (x)h K1 (x, t) = K2 (x, t) =

k=1 n

rk (x)sk (t),

˜ 2 (x, t) = K

k=1

k=1 n

(11.53) r˜k (x)˜ sk (t).

k=1

The direct computation method can be applied as follows: 1. We ﬁrst substitute (11.53) into the system of Fredholm integro-diﬀerential equations (11.52) to obtain b b n n ˜hk (t)v(t)dt, gk (x) hk (t)u(t)dt + g˜k (x) u(i) (x) = f1 (x) + v (i) (x) = f2 (x) +

k=1 n k=1

a b

rk (x)

sk (t)u(t)dt + a

k=1 n k=1

a b

r˜k (x)

s˜k (t)v(t)dt. a

(11.54) 2. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (11.54) becomes u(i) (x) = f1 (x) + α1 g1 (x) + · · · + αn gn (x) + β1 g˜1 (x) + · · · + βn g˜n (x), v (i) (x) = f2 (x) + γ1 r1 (x) + · · · + γn rn (x) + δ1 r˜1 (x) + · · · + δn r˜n (x), (11.55) where

354

11 Systems of Fredholm Integral Equations

b

hi (t) u(t)dt, 1 i n,

αi = a

b

si (t) u(t)dt, 1 i n,

γi =

b

βi =

(11.56)

b

s˜i (t) v(t)dt, 1 i n.

δi =

a

˜hi (t) v(t)dt, 1 i n,

a

a

3. Integrating both sides of (11.55) i times from 0 to x, using the given initial conditions, and substituting the resulting equations for u(x) and v(x) into (11.56) gives a system of algebraic equations that can be solved to determine the constants αi , βi , γi , and δi . Using the obtained numerical values of these constants into the obtained equations for u(x) and v(x), the solutions u(x) and v(x) of the system of Fredholm integro-diﬀerential equations (11.52) follow immediately. Example 11.9 Solve the following system of Fredholm integro-diﬀerential equations by using the direct computation method π u (x) = sin x + x cos x + (2 − π2 ) + (tu(t) − v(t)) dt, u(0) = 0, 0 (11.57) π (u(t) − tv(t)) dt, v(0) = 0. v (x) = cos x − x sin x − 3π + 0

Following the analysis presented above, this system can be rewritten as u (x) = sin x + x cos x + (2 − π2 + α), (11.58) v (x) = cos x − x sin x + (β − 3π), where π

α= 0

π

(tu(t) − v(t))dt,

β= 0

(u(t) − tv(t))dt.

(11.59)

Integrating both sides of (11.58) once from 0 to x, and using the initial conditions we ﬁnd u(x) = x sin x + (2 − π 2 + α)x, (11.60) v(x) = x cos x + (β − 3π)x. To determine α, and β, we substitute (11.60) into (11.59) and solving the resulting system we obtain (11.61) α = π2 − 2, β = 3π. Substituting (11.61) into (11.60) leads to the exact solutions (u(x), v(x)) = (x sin x, x cos x).

(11.62)

Example 11.10 Solve the following system of Fredholm integro-diﬀerential equations by using the direct computation method

11.3 Systems of Fredholm Integro-Diﬀerential Equations

u (x) = −1 + sinh x +

ln 2

(u(t) + v(t))dt, u(0) = 1, 0

v (x) = 1 − 2 ln 2 + cosh x +

(11.63)

ln 2

(tu(t) + tv(t))dt, v(0) = 0. 0

This system can be rewritten as u (x) = sinh x + α − 1, v (x) = cosh x + 1 − 2 ln 2 + β, where ln 2 ln 2 (u(t) + v(t))dt, β = (tu(t) + tv(t))dt. α= 0

355

(11.64)

(11.65)

0

Integrating both sides of (11.64) once from 0 to x, and using the initial conditions we ﬁnd u(x) = cosh x + (α − 1)x, (11.66) v(x) = sinh x + (1 − 2 ln 2 + β)x. To determine α, and β, we substitute (11.66) into (11.65) and solving the resulting system we obtain α = 1, β = 2 ln 2 − 1. (11.67) Substituting (11.67) into (11.66) leads to the exact solutions (u(x), v(x)) = (cosh x, sinh x).

(11.68)

Example 11.11 Solve the following system of Fredholm integro-diﬀerential equations by using the direct computation method ln 2 5 x (u(t) + v(t))dt, u(0) = 1, u (x) = e − + 2 0 (11.69) ln 2 1 v (x) = 2e2x + + (tu(t) − tv(t))dt, v(0) = 1. 4 0 This system can be rewritten as 5 1 (11.70) u (x) = ex + α − , v (x) = 2e2x + + β, 2 4 where ln 2 ln 2 (11.71) α= (u(t) + v(t))dt, β = (tu(t) − tv(t))dt. 0

0

Integrating both sides of (11.70) once from 0 to x, and using the initial conditions we ﬁnd 5 1 x, v(x) = e2x + + β x. (11.72) u(x) = ex + α − 2 4 To determine α, and β, we substitute (11.72) into (11.71) and solving the resulting system we obtain

356

11 Systems of Fredholm Integral Equations

1 5 , β=− . 2 4 Substituting (11.73) into (11.72) leads to the exact solutions α=

(u(x), v(x)) = (ex , e2x ).

(11.73) (11.74)

Example 11.12 Solve the following system of Fredholm integro-diﬀerential equations by using the direct computation method π2 π ((x − t)u(t) − (x − t)v(t)) dt, u (x) = − cos x − 2 − + 2 0 u(0) = 1, u (0) = 0 π2 π ((x + t)u(t) − (x + t)v(t)) dt, v (x) = − sin x + 2 − + 2 0

(11.75)

v(0) = 0, v (0) = 1. This system can be rewritten as

π u (x) = − cos x − 2 − + αx − β, 2 π + αx + β, v (x) = − sin x + 2 − 2

where

π 2

α= 0

(u(t) − v(t))dt,

π 2

β= 0

(tu(t) − tv(t))dt.

(11.76)

(11.77)

Integrating both sides of (11.76) twice from 0 to x, and using the initial conditions and proceeding as before we obtain π α = 0, β = − 2. (11.78) 2 This in turn gives the exact solutions (u(x), v(x)) = (cos x, sin x).

(11.79)

Exercises 11.3.1 Use the direct computation method to solve the following systems of Fredholm integro-diﬀerential equations ⎧ π ⎪ ⎪ (u(t) − tv(t)) dt, u(0) = 0 ⎨ u (x) = cos x − 4 + 0 1. π ⎪ ⎪ (tu(t) − v(t)) dt, v(0) = 1 ⎩ v (x) = − sin x − π + 0

11.3 Systems of Fredholm Integro-Diﬀerential Equations π ⎧ π 2 ⎪ ⎪ (x) = − sin x − 2x + ((x − t)u(t) − (x − t)v(t)) dt + u ⎪ ⎪ 2 ⎪ 0 ⎪ ⎨ π π 2. 2 ⎪ ((x + t)u(t) − (x + t)v(t)) dt v (x) = − cos x − 2x − + ⎪ ⎪ 2 ⎪ 0 ⎪ ⎪ ⎩ u(0) = 2, v(0) = 1 π ⎧ π ⎪ ((x − t)u(t) − (x + t)v(t)) dt ⎪ u (x) = −2 sin(2x) − + ⎪ ⎪ 2 ⎪ 0 ⎨ π π 3. ⎪ ((x + t)u(t) − (x − t)v(t)) dt v (x) = 2 cos(2x) + + ⎪ ⎪ 2 ⎪ 0 ⎪ ⎩ u(0) = 1, v(0) = 0 ⎧ ln 2 1 ⎪ 2 ⎪ (ln 2) u (x) = sinh 2x + + (tu(t) − tv(t)) dt ⎪ ⎪ ⎪ 2 ⎪ 0 ⎨ ln 2 4. 15 ⎪ − 2 ln 2 + v (x) = sinh 2x − (u(t) + v(t)) dt ⎪ ⎪ ⎪ 16 0 ⎪ ⎪ ⎩ u(0) = 1, v(0) = 2 ⎧ ln 2 ⎪ 2 ⎪ (x) = sinh 2x − (ln 2) + (tu(t) + tv(t)) dt, u(0) = 2 u ⎪ ⎨ 5.

0

⎪ ⎪ ⎪ ⎩ v (x) = −2 sinh 2x − 2 ln 2 +

ln 2 0

(u(t) + v(t)) dt, v(0) = 0

⎧ ln 2 3 1 ⎪ ⎪ u (x) = cosh x + (tu(t) − tv(t)) dt, u(0) = 0 − ln 2 + ⎪ ⎨ 2 2 0 6. ln 2 ⎪ ⎪ ⎪ ⎩ v (x) = sinh x − (ln 2)2 + (u(t) + v(t)) dt, v(0) = 1 0

⎧ ln 2 2 ⎪ x 3 ⎪ (tu(t) + tv(t)) dt, u(0) = 1 ⎪ ⎨ u (x) = 1 + e − 3 (ln 2) + 0 7. ln 2 ⎪ ⎪ ⎪ ⎩ v (x) = 1 − ex − (ln 2)2 + (u(t) + v(t)) dt, v(0) = −1 0

ln 2 ⎧ 1 ⎪ x ⎪ (1 − 3 ln 2) + (x) = (1 + x)e + (u(t) + v(t)) dt u ⎪ ⎪ 2 ⎪ 0 ⎪ ⎨ ln 2 1 8. ⎪ v (x) = (1 − x)e−x + (3 − 5 ln 2) + (u(t) − v(t)) dt ⎪ ⎪ 2 ⎪ 0 ⎪ ⎪ ⎩ u(0) = 0, v(0) = 0 ⎧ ln 2 1 ⎪ x ⎪ + (x) = e − 2 ln 2 − (tu(t) + v(t)) dt, u(0) = 1 u ⎪ ⎨ 2 0 9. ln 2 ⎪ 1 ⎪ ⎪ ⎩ v (x) = 2e2x − 2 ln 2 − + (u(t) + tv(t)) dt, v(0) = 1 4 0

357

358

11 Systems of Fredholm Integral Equations ⎧ 4 ⎪ u (x) = 2 cos 2x + cos x ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪ π ⎪ ⎪ ⎪ 2 ⎪ ⎪ + (sin(x − t)u(t) − cos(x − t)v(t)) dt, u(0) = 0, u (0) = 0 ⎨

10.

0

⎪ 2 ⎪ ⎪ ⎪ ⎪ v (x) = −2 cos 2x − cos x ⎪ 3 ⎪ ⎪ ⎪ π ⎪ ⎪ 2 ⎪ ⎩ + (cos(x − t)u(t) − sin(x − t)v(t)) dt, v(0) = 1, v (0) = 0 0

π ⎧ ⎪ (sin(x − t)u(t) − cos(x − t)v(t)) dt, u (x) = − sin x + π cos x + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎨ u(0) = 0, u (0) = 1 π 11. ⎪ ⎪ v (x) = − cos x − π sin x + ⎪ (cos(x − t)u(t) + sin(x − t)v(t)) dt, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩ v(0) = 1, v (0) = 0 ⎧ u (x) = −(1 + π) sin x − cos x ⎪ ⎪ π ⎪ ⎪ ⎪ ⎪ + (cos(x − t)u(t) + cos(x − t)v(t)) dt, u(0) = 1, u (0) = 1 ⎪ ⎨ 12.

0

⎪ v (x) = (1 − π) cos x − sin x ⎪ ⎪ ⎪ π ⎪ ⎪ ⎪ ⎩ (cos(x − t)u(t) − cos(x − t)v(t)) dt, v(0) = −1, v (0) = 1 + 0

11.3.2 The Variational Iteration Method In Chapters 4 and 6, the variational iteration method was used to handle the Fredholm integral equations and the Fredholm integro-diﬀerential equations respectively. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method [8] handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the so called Adomian polynomials that we need for nonlinear problems. The correction functionals for the system integro-diﬀerential equations (11.52) are given by un+1 (x) = un (x) b x (i) ˜ λ(t) un (t) − f1 (t) − un (r)dr + K1 (t, r)˜ vn (r)dr dt, + K1 (t, r)˜ 0

a

vn+1 (x) = vn (x) x λ(t) u(i) (t) − f (t) − + 2 n 0

a

b

(11.80) ˜ K2 (t, r)˜ un (r)dr + K2 (t, r)˜ vn (r)dr dt.

11.3 Systems of Fredholm Integro-Diﬀerential Equations

359

It is required ﬁrst to determine the Lagrange multiplier λ that can be identiﬁed optimally. Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), vn+1 (x), n 0 of the solutions u(x) and v(x). The zeroth approximations u0 (x) and v0 (x) can be any selective functions. However, we can use the initial conditions to select the zeroth approximations u0 and v0 (x). The VIM will be illustrated by studying the following examples. Example 11.13 Solve the system of Fredholm integro-diﬀerential equations by using the variational iteration method π

u (x) = −2 − sin x +

(u(t) + v(t))dt, u(0) = 1, 0

v (x) = 2 − π + cos x +

(11.81)

π

(tu(t) + tv(t))dt, v(0) = 0. 0

The correction functionals for this system are given by x un+1 (x) = un (x) − (un (t) + 2 + sin t − ρ1 ) dt, vn+1 (x) = vn (x) − where ρ1 =

0

0 x

0

(11.82) (vn (t)

+ π − 2 − cos t − ρ2 ) dt,

π

(un (r) + vn (r))dr,

ρ2 =

0

π

(run (r) + rvn (r))dr.

(11.83)

Selecting u0 (x) = 1 and v0 (x) = 0, the correction functionals gives the following successive approximations u0 (x) = 1, v0 (x) = 0, u1 (x) = u0 (x) −

0

x

u0 (t)

+ 2 + sin t −

0

π

(u0 (r) + v0 (r))dr dt,

= cos x − 2x + πx, x π (ru0 (r) + rv0 (r))dr dt, v0 (t) + π − 2 − cos t − v1 (x) = v0 (x) − 0

0

2

π = sin x + 2x − πx + x, (11.84) 2 x π (u1 (r) + v1 (r))dr dt, u1 (t) + 2 + sin t − u2 (x) = u1 (x) − 0

0

= cos x + (2x − 2x) + (πx − πx) + · · · , x π v1 (t) + π − 2 − cos t − v2 (x) = v1 (x) − (ru1 (r) + rv1 (r))dr dt, 0

0

360

11 Systems of Fredholm Integral Equations

= sin x + (2x − 2x) + (πx − πx) +

π2 π2 x− x + ··· , 2 2

and so on. It is obvious that noise terms appear in each component. By canceling the noise terms, the exact solutions are therefore given by (u(x), v(x)) = (cos x, sin x).

(11.85)

Example 11.14 Solve the system of Fredholm integro-diﬀerential equations by using the variational iteration method ln 2 (u(t) + v(t))dt, u(0) = 0, u (x) = −1 + cosh x + 0 (11.86) v (x) = 1 − 2 ln 2 + sinh x +

ln 2

(tu(t) + tv(t))dt, v(0) = 1. 0

The correction functionals for this system are given by x (un (t) + 1 − cosh t − ρ3 ) dt, un+1 (x) = un (x) − vn+1 (x) = vn (x) −

0 x

0

where

(11.87) (vn (t)

ρ3 = ρ4 =

+ 2 ln 2 − 1 − sinh t − ρ4 ) dt.

ln 2

(un (r) + vn (r))dr,

0

(11.88)

ln 2 0

(run (r) + rvn (r))dr.

We can use the initial conditions to select u0 (x) = 0 and v0 (x) = 1. Using this selection into the correction functionals gives the following successive approximations u0 (x) = 0, v0 (x) = 1, u1 (x) = sinh x + x ln 2 − x, v1 (x) = cosh x + x − 2x ln 2 +

x (ln 2)2 , 2

x

(11.89)

x (ln 2)2 + · · · , 2 2 x x v2 (x) = cosh x + (x − x) + (2x ln 2 − 2x ln 2) + (ln 2)2 − (ln 2)2 + · · · , 2 2 x x 3 3 u3 (x) = sinh x + (ln 2) − (ln 2) + · · · , 4 4 x x (ln 2)4 − (ln 2)4 + · · · , v3 (x) = cosh x + 3 3

u2 (x) = sinh x + (x − x) + (2x ln 2 − 2x ln 2) +

(ln 2)2 −

11.3 Systems of Fredholm Integro-Diﬀerential Equations

361

and so on. It is obvious that noise terms appear in each component. By canceling the noise terms, the exact solutions are therefore given by (u(x), v(x)) = (sinh x, cosh x). (11.90) Example 11.15 Solve the system of Fredholm integro-diﬀerential equations by using the variational iteration method ln 3 u (x) = ex − 6 + (u(t) + v(t))dt, u(0) = 1, 0 (11.91) ln 3

v (x) = 2e2x + 2 +

0

(u(t) − v(t))dt, v(0) = 1.

The correction functionals for this system are given by x ln 3 un (t) − et + 6 − (un (r) + vn (r))dr dt, un+1 (x) = un (x) − vn+1 (x) = vn (x) −

0

x 0

0

vn (t) − 2e2t − 2 −

0

ln 3

(un (r) − vn (r))dr

dt,

(11.92) where we used the Lagrange multiplier λ = −1 for the each of ﬁrst order Fredholm integro-diﬀerential equations. Selecting u0 (x) = 1 and v0 (x) = 1gives the following approximations u0 (x) = 1, v0 (x) = 1, u1 (x) = ex − 6x + 2x ln 3, v1 (x) = e2x + 2x, u2 (x) = ex + (6x − 6x) + (2x ln 3 − 2x ln 3) − 2x(ln 3)2 + x(ln 3)3 , (11.93) v2 (x) = e2x + (2x − 2x) − 4x(ln 3)2 + x(ln 3)3 , u3 (x) = ex + (2x(ln 3)2 − 2x(ln 3)2 ) + (x(ln 3)3 − x(ln 3)3 ) + · · · , v3 (x) = e2x + (4x(ln 3)2 − 4x(ln 3)2 ) + (x(ln 3)3 − x(ln 3)3 ) + · · · , and so on. It is obvious that noise terms appear in each component. By canceling the noise terms, the exact solutions are therefore given by (11.94) (u(x), v(x)) = (ex , e2x ). Example 11.16 Solve the system of Fredholm integro-diﬀerential equations by using the variational iteration method π 3π (2 + π) + u (x) = 2 cos 2x − (u(t) + tv(t))dt, u(0) = 1, u (0) = 0 4 0 π 3π (2 − π) + (tu(t) − v(t))dt, v(0) = 2, v (0) = 0. v (x) = −2 cos 2x + 4 0 (11.95)

362

11 Systems of Fredholm Integral Equations

The correction functionals for this system are given by x 3π un+1 (x) = un (x) + (t − x)(un (t) − 2 cos 2t + (2 + π) − ρ5 ) dt, 4 0 x 3π vn+1 (x) = vn (x) + (t − x)(vn (t) + 2 cos 2t − (2 − π) − ρ6 ) dt. 4 0 (11.96) where π

ρ5 =

0

π

(un (r) + rvn (r))dr,

ρ6 =

0

(run (r) − vn (r))dr.

(11.97)

Notice that the Lagrange multiplier λ = (t − x) because each equation is of second order. We can use the initial conditions to select u0 (x) = 1 and v0 (x) = 2. Using this selection into the correction functionals and proceeding as before we obtain the following successive approximations u0 (x) = 1, v0 (x) = 2, π u1 (x) = 1 + sin2 x − (2 − π)x2 , 8 π 2 (11.98) v1 (x) = 1 + cos x − (2 + π)x2 , 8 π π u2 (x) = 1 + sin2 x + (2 − π)x2 − (2 − π)x2 + · · · , 8 8 π π (2 + π)x2 − (2 + π)x2 + · · · , v2 (x) = 1 + cos2 x + 8 8 and so on. It is obvious that noise terms appear as explained in the previous examples. By canceling the noise terms, the exact solutions are therefore given by (u(x), v(x)) = (1 + sin2 x, 1 + cos2 x). (11.99)

Exercises 11.3.2 Use the variational iteration method to solve the following systems of Fredholm integro-diﬀerential equations ⎧ π ⎪ ⎪ (x) = cos x − 4 + (u(t) − tv(t)) dt, u(0) = 0 u ⎪ ⎨ 1.

0

⎪ ⎪ ⎪ ⎩ v (x) = − sin x − π +

π

0

(tu(t) − v(t)) dt, v(0) = 1

⎧ π π 2 ⎪ (x) = cos x − 2x + ⎪ + ((x − t)u(t) + (x − t)v(t)) dt, u(0) = 0 u ⎪ ⎨ 2 0 2. π ⎪ π ⎪ 2 ⎪ ⎩ v (x) = − sin x − 2 + + ((x + t)u(t) − (x + t)v(t)) dt, v(0) = 1 2 0

11.3 Systems of Fredholm Integro-Diﬀerential Equations ⎧ π ⎪ 2 ⎪ (x) = sin x + x cos x + (2 − π ) + (tu(t) − v(t)) dt, u(0) = 0 u ⎪ ⎨ 0 3. π ⎪ ⎪ ⎪ (u(t) − tv(t)) dt, v(0) = 0 ⎩ v (x) = cos x − x sin x − 3π + 0

⎧ ln 2 1 ⎪ 2 ⎪ (ln 2) (x) = sinh 2x + + (tu(t) − tv(t)) dt, u(0) = 1 u ⎪ ⎨ 2 0 4. ln 2 ⎪ 15 ⎪ ⎪ ⎩ v (x) = sinh 2x − − 2 ln 2 + (u(t) + v(t)) dt, v(0) = 2 16 0 ⎧ ln 2 ⎪ 2 ⎪ u (x) = sinh 2x − (ln 2) + (tu(t) + tv(t)) dt, u(0) = 1 ⎪ ⎨ 5.

0

⎪ ⎪ ⎪ ⎩ v (x) = − sinh 2x − 2 ln 2 +

ln 2

0

(u(t) + v(t)) dt, v(0) = 1

⎧ ln 2 3 1 ⎪ ⎪ (x) = cosh x + (tu(t) − tv(t)) dt, u(0) = 0 − ln 2 + u ⎪ ⎨ 2 2 0 6. ln 2 ⎪ ⎪ ⎪ ⎩ v (x) = sinh x − (ln 2)2 + (u(t) + v(t)) dt, v(0) = 1 0

⎧ ln 2 3 5 ⎪ x ⎪ u (x) = e + (tu(t) − tv(t)) dt, u(0) = 1 − ln 2 + ⎪ ⎨ 2 2 0 7. ln 2 ⎪ 3 ⎪ ⎪ ⎩ v (x) = −e−x − + (u(t) + v(t)) dt, v(0) = 1 2 0 ⎧ ln 2 1 ⎪ x ⎪ u (x) = (1 + x)e + (u(t) + v(t)) dt, u(0) = 0 (1 − 3 ln 2) + ⎪ ⎨ 2 0 8. ln 2 ⎪ 1 ⎪ ⎪ ⎩ v (x) = (1 − x)e−x + (3 − 5 ln 2) + (u(t) − v(t)) dt, v(0) = 0 2 0 ⎧ ln 2 10 ⎪ x ⎪ + (x) = e − (u(t) + v(t)) dt, u(0) = 1 u ⎪ ⎨ 3 0 9. ln 2 ⎪ 4 ⎪ ⎪ ⎩ v (x) = 3e3x + + (u(t) − v(t)) dt, v(0) = 1 3 0 ⎧ ln 2 10 ⎪ ⎪ + (u(t) + v(t)) dt, u(0) = 1, u (0) = 1 ⎪ u (x) = ex − ⎨ 3 0 10. ln 2 ⎪ 4 ⎪ ⎪ ⎩ v (x) = 9e3x + + (u(t) − v(t)) dt, v(0) = 1, v (0) = 3 3 0 π ⎧ ⎪ ⎪ (x) = − sin x + π cos x + (sin(x − t)u(t) − cos(x − t)v(t)) dt, u ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u(0) = 0, u (0) = 1 11. π ⎪ ⎪ ⎪ ⎪ (x) = − cos x − π sin x + (cos(x − t)u(t) + sin(x − t)v(t)) dt, v ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ v(0) = 1, v (0) = 0

363

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11 Systems of Fredholm Integral Equations

⎧ π ⎪ (cos(x − t)u(t) + cos(x − t)v(t)) dt, u (x) = −(1 + π) sin x − cos x + ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ u(0) = 1, u (0) = 1 π 12. ⎪ ⎪ ⎪ (cos(x − t)u(t) − cos(x − t)v(t)) dt, v (x) = (1 − π) cos x − sin x + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩ v(0) = −1, v (0) = 1

References 1. R.P. Agarwal, D. O’Regan and P. Wong, Eigenvalues of a system of Fredholm integral equations, Math. Comput. Modelling, 39 (2004) 1113–1150. 2. R.F. Churchhouse, Handbook of Applicable Mathematics, Wiley, New York, (1981). 3. R.P. Kanwal, Linear Integral Equations, Birkhauser, Boston, (1997). 4. K. Maleknejad, N. Aghazadeh and M. Rabbani, Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method, Appl. Math. Comput., 175 (2006) 1229-1234. 5. A. Polyanin and A. Manzhirov, Handbook of Integral Equations, Chapman and Hall, (2008). 6. G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994). 7. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 8. A.M. Wazwaz, The variational iteration method for solving linear and nonlinear systems of PDEs, Comput. Math. Appl., 54 (2007) 895–902.

Chapter 12

Systems of Singular Integral Equations

12.1 Introduction Systems of singular integral equations appear in many branches of scientiﬁc ﬁelds [1–6], such as microscopy, seismology, radio astronomy, electron emission, atomic scattering, radar ranging, plasma diagnostics, X-ray radiography, and optical ﬁber evaluation. Studies of systems of singular integral equations have attracted much concern in applied sciences. The use of computer symbolic systems such as Maple and Mathematica facilitates the tedious work of computation. The general ideas and the essential features of these systems are of wide applicability. The well known systems of singular integral equations [7] are given by x f1 (x) = (K11 (x, t)u(t) + K12 (x, t)v(t)) dt, 0 (12.1) x (K21 (x, t)u(t) + K22 (x, t)v(t)) dt. f2 (x) = 0

and u(x) = f1 (x) +

v(x) = f2 (x) +

x

(K11 (x, t)u(t) + K12 (x, t)v(t)) dt,

0

(12.2)

x 0

(K21 (x, t)u(t) + K22 (x, t)v(t)) dt,

where the kernels Kij are singular kernels given by 1 , 1 i, j 2. (12.3) Kij = (x − t)αij The system (12.1) and the system (12.2) are called the system of the generalized Abel singular integral equations and the system of the weakly generalized singular integral equations respectively. For αij = 12 , the system (12.1) is called the system of Abel singular integral equations.

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

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12 Systems of Singular Integral Equations

12.2 Systems of Generalized Abel Integral Equations In Chapter 7, the singular integral equations were presented. The Abel’s singular integral equation and its generalized form, and the weakly singular integral equations were handled by a variety of methods. In this section, the systems of Abel’s generalized singular integral equations will be examined by using only the Laplace transform method that we used before in Chapter 7 among other places. In this section, ﬁrst we will study systems of two unknown functions u(x) and v(x), and then we will study systems of three unknown functions u(x), v(x) and w(x) as well. Generalization to any number of unknowns can be easily developed.

12.2.1 Systems of Generalized Abel Integral Equations in Two Unknowns The system of generalized Abel integral equation in two unknowns is of the form x (K11 (x, t)u(t) + K12 (x, t)v(t)) dt, f1 (x) = 0 (12.4) x f2 (x) = (K21 (x, t)u(t) + K22 (x, t)v(t)) dt. 0

The kernels Kij (x, t), 1 i, j 2 and the functions fi (x), i = 1, 2 are given real-valued functions. Recall that the kernels Kij are singular kernels given by 1 , 1 i, j 2. (12.5) Kij = (x − t)αij For αij = 12 , 1 i, j 2, the system is called system of Abel integral equations. Abel’s systems of three equations in three unknowns will be examined in details in the next section. The system of Abel’s generalized singular integral equations (12.4) gives a solution if K11 K12 = 0. (12.6) det K21 K22 In what follows we will apply the Laplace transform method to handle the system (12.4). Taking Laplace transform of both sides of the system (12.4) gives the linear system in U (s) and V (s) F1 (s) = K11 (s)U (s) + K12 (s)V (s), (12.7) F2 (s) = K21 (s)U (s) + K22 (s)V (s), where

12.2 Systems of Generalized Abel Integral Equations

U (s) = L{u(x)}, Fi (s) = L{fi (x)}, Kij (s) = L{Kij (x)},

367

V (s) = L{v(x)}, 1 i 2, 1 i, j 2.

(12.8)

Solving (12.7) for U (s) and V (s) by using Cramer’s rule gives F1 (s) F2 (s)

K12 (s) K22 (s) , U (s) = K11 (s) K12 (s) K12 (s) K22 (s)

K11 (s) K21 (s)

F1 (s) F2 (s) . V (s) = K11 (s) K12 (s) K21 (s) K22 (s)

(12.9)

Having determined U (s) and V (s), the unique solution for u(x) and v(x) can be determined by using the inverse Laplace transform method. Table 1 in Section 1.5 shows the Laplace transforms and the inverse Laplace transforms for a variety of functions. It is interesting to point out that the computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations. This will be used here to facilitate the computational work. Moreover, recall that the linear system (12.7) has a unique solution for u(x) and v(x) if and only if K11 (s) K12 (s) (12.10) K12 (s) K22 (s) = 0. It is worth noting that for simplicity reasons we will focus our study only on the speciﬁc case where α11 = α22 , α12 = α21 . (12.11) The other cases of αij can be handled by a like manner. In this case, the use of the computer symbolic systems is necessary to perform tedious calculations. Example 12.1 Solve the system of singular integral equations by using the Laplace transform method x √ 3 2 2 1 3 1 3 dt, 2 x 1+ x + x 1− x = 1 u(t) + 1 v(t) 3 2 5 (x − t) 2 (x − t) 3 0 x √ 3 2 2 1 3 1 u(t) + v(t) dt. 2 x 1 − x + x3 1 + x = 1 1 3 2 5 (x − t) 3 (x − t) 2 0 (12.12) Taking Laplace transform of both sides of each equation in (12.12) gives √ −5 √ 2 2 −2 8 1 s− 3 (1 − s) = πs− 2 U (s) + Γ s 3 V (s), πs 2 (1 + s) − Γ 3 3 √ √ 2 2 5 8 2 1 s− 3 (1 + s) = Γ s− 3 U (s) + πs− 2 V (s). − πs− 2 (1 − s) + Γ 3 3 (12.13) Solving this systems of equations for U (s) and V (s), by using any computer symbolic system, such as Maple or Mathematica, gives 1 1 1 1 (12.14) U (s) = + 2 , V (s) = − 2 . s s s s

368

12 Systems of Singular Integral Equations

By taking the inverse Laplace transform of both sides of each equation in (12.14), the exact solutions are given by (u(x), v(x)) = (1 + x, 1 − x). (12.15) Example 12.2 Solve the system of singular integral equations by using the Laplace transform method 16 5 25 9 x 4 (32x2 + 39) + x 5 (57x2 − 133) 195 4788 x 1 1 dt, = 3 u(t) + 1 v(t) (x − t) 4 (x − t) 5 0 (12.16) 16 5 25 9 x 4 (32x2 − 39) + x 5 (57x2 + 133) 195 4788 x 1 1 u(t) + v(t) dt. = 1 3 (x − t) 5 (x − t) 4 0 Taking Laplace transform of both sides of each equation in (12.16) gives

√ √ Γ 45 Γ 45 π 2 π 2 6 6 + 1 + 14 − 1 = 1 3 U (s) + V (s),

9 4 s2 s2 s 4 Γ 34 s5 s4 Γ 4 s5

√ √ Γ 45 Γ 45 π 2 6 6 π 2 − 1 + 14 +1 = U (s) + 1 3 V (s).

9 4 s2 s2 s 4 Γ 34 s5 s5 s4 Γ 4 (12.17) Solving this systems of equations for U (s) and V (s), by using any computer symbolic system, such as Maple or Mathematica, gives 1 6 1 6 (12.18) U (s) = 4 + 2 , V (s) = 4 − 2 . s s s s By taking the inverse Laplace transform of both sides of each equation in (12.18), the exact solutions are given by (12.19) (u(x), v(x)) = (x3 + x, x3 − x).

Exercises 12.2.1 Solve the system of singular integral equations by using the Laplace transform method ⎧ x 5 4 1 1 8 1 ⎪ ⎪ ⎪ dt x 4 (6x + 5) + x 5 (20x + 27) = 3 u(t) + 1 v(t) ⎪ ⎨5 36 0 (x − t) 4 (x − t) 5 1. x ⎪ ⎪4 1 5 4 1 1 ⎪ ⎪ 4 5 x x (16x + 15) + (5x + 6) = u(t) + v(t) dt ⎩ 1 3 5 12 0 (x − t) 5 (x − t) 4

12.2 Systems of Generalized Abel Integral Equations 369 ⎧ x 1 5 4 3 2 1 ⎪ ⎪ 5 (5x + 9π) + 3 (−3x + 5π) = ⎪ x x dt 1 u(t) + 1 v(t) ⎪ ⎨ 36 10 0 (x − t) 5 (x − t) 3 2. x ⎪ ⎪ 3 2 1 1 5 4 ⎪ ⎪ 5 3 (−5x + 9π) + (3x + 5π) = u(t) + v(t) dt x x ⎩ 1 1 36 10 0 (x − t) 3 (x − t) 5 ⎧ x 5 4 1 4 1 1 ⎪ ⎪ ⎪ x 2 (x + 9) + x 5 (5x − 54) = dt 1 u(t) + 1 v(t) ⎪ ⎨3 36 0 (x − t) 2 (x − t) 5 3. x ⎪ ⎪ 1 5 4 1 4 1 ⎪ ⎪ 2 5 (x − 9) + (5x + 54) = u(t) + v(t) dt x x ⎩ 1 1 3 36 0 (x − t) 5 (x − t) 2 ⎧ x 1 256 9 1 16 3 ⎪ ⎪ 4 + 2 = ⎪ u(t) + v(t) dt x x 1 1 ⎪ ⎨ 3 315 0 (x − t) 2 (x − t) 4 4. x ⎪ ⎪ 1 8192 19 1 1 ⎪ ⎪ 4 2 x + 8x = u(t) + v(t) dt ⎩ 1 1 21945 0 (x − t) 4 (x − t) 2 ⎧ x 1 1 ⎪ ⎪ 2 x 12 (8x2 + 15) − 3 x 23 (9x2 − 20) = ⎪ u(t) + v(t) dt 1 1 ⎪ ⎨ 15 40 0 (x − t) 2 (x − t) 3 5. x ⎪ ⎪ 3 2 1 1 2 1 ⎪ 2 2 ⎪ 2 3 dt x (9x + 20) = ⎩ − x (8x − 15) + 1 u(t) + 1 v(t) 15 40 0 (x − t) 3 (x − t) 2 ⎧ x 1 1 ⎪ ⎪ 2 x 12 (16x3 + 35) + 3 x 23 (81x3 − 220) = ⎪ u(t) + v(t) dt 1 1 ⎪ ⎨ 35 440 0 (x − t) 2 (x − t) 3 6. x ⎪ ⎪ 1 2 1 3 2 1 ⎪ 3 3 ⎪ 2 3 x (16x − 35) + x (81x + 220) = dt ⎩ 1 u(t) + 1 v(t) 35 440 0 (x − t) 3 (x − t) 2 ⎧ 3 2 2 1 ⎪ x 3 (9x2 − 12x + 20) ⎪ − x 2 (8x2 − 10x − 15) + ⎪ ⎪ 15 40 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ dt = 1 u(t) + 1 v(t) ⎪ ⎨ 0 (x − t) 2 (x − t) 3 7. 2 1 3 2 ⎪ ⎪ ⎪ x 2 (8x2 − 10x + 15) − x 3 (9x2 − 12x − 20) ⎪ ⎪ 15 40 ⎪ ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ = u(t) + v(t) dt ⎩ 1 1 0 (x − t) 3 (x − t) 2 ⎧ 3 3 2 4 ⎪ x 4 (128x3 + 220x − 385) + x 3 (81x3 + 132x + 220) ⎪ − ⎪ ⎪ 1155 440 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ dt = 1 u(t) + 1 v(t) ⎪ ⎨ 0 (x − t) 3 (x − t) 4 8. 3 4 3 2 ⎪ ⎪ ⎪ x 4 (128x3 + 220x + 385) − x 3 (81x3 + 132x − 220) ⎪ ⎪ 1155 440 ⎪ ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ u(t) + v(t) dt = ⎩ 1 1 0 (x − t) 4 (x − t) 3

370

12 Systems of Singular Integral Equations

12.2.2 Systems of Generalized Abel Integral Equations in Three Unknowns The system of Abel’s generalized singular integral equations in three unknowns is of the form x

f1 (x) = f2 (x) = f3 (x) =

(K11 (x, t)u(t) + K12 (x, t)v(t) + K13 (x, t)w(t)) dt,

0 x

(K21 (x, t)u(t) + K22 (x, t)v(t) + K23 (x, t)w(t)) dt,

0

(12.20)

x 0

(K31 (x, t)u(t) + K32 (x, t)v(t) + K33 (x, t)w(t)) dt.

The kernels Kij (x, t), 1 i, j 3 and the functions fi (x), i = 1, 2, 3 are given real-valued functions. Recall that the kernels Kij are singular kernels given by 1 Kij = , 1 i, j 3. (12.21) (x − t)αij For αij = 12 , 1 i, j 3, the system is called system of Abel integral equations in three unknowns. The system of Abel’s generalized singular integral equations in three unknowns gives a solution if ⎤⎞ ⎛⎡ K11 K12 K13 (12.22) det ⎝⎣ K21 K22 K23 ⎦⎠ = 0. K31 K32 K33 In what follows we will apply the Laplace transform method to handle the system (12.20). Taking Laplace transform of both sides of the system (12.20) gives the linear system in U (s), V (s), and W (s) F1 (s) = K11 (s)U (s) + K12 (s)V (s) + K13 (s)W (s), F2 (s) = K21 (s)U (s) + K22 (s)V (s) + K23 (s)W (s), (12.23) F3 (s) = K31 (s)U (s) + K32 (s)V (s) + K33 (s)W (s). where U (s) = L{u(x)}, V (s) = L{v(x)}, W (s) = L{w(x)}, (12.24) Fi (s) = L{fi (x)}, 1 i 3, Kij (s) = L{Kij (x)}, 1 i, j 3, and the singular kernels Kij are given above in (12.21). Solving (12.23) for U (s), V (s), and W (s) by using Cramer’s rule gives F (s) K (s) K (s) 1 12 13 F (s) K (s) K (s) 22 23 2 F3 (s) K32 (s) K33 (s) , (12.25) U (s) = K11 (s) K12 (s) K13 (s) K (s) K (s) K (s) 22 23 21 K31 (s) K32 (s) K33 (s)

12.2 Systems of Generalized Abel Integral Equations

371

K (s) 11 K (s) 21 K31 (s)

K (s) 11 K (s) 21 K31 (s)

F1 (s) K13 (s) F2 (s) K23 (s) F3 (s) K33 (s) , V (s) = K11 (s) K12 (s) K13 (s) K (s) K (s) K (s) 22 23 21 K31 (s) K32 (s) K33 (s) and

K12 (s) F1 (s) K22 (s) F2 (s) K23 (s) F3 (s) . W (s) = K11 (s) K12 (s) K13 (s) K (s) K (s) K (s) 22 23 21 K31 (s) K32 (s) K33 (s)

(12.26)

(12.27)

The linear system (12.23) gives a unique solution for u(x), v(x) and w(x) if and only if K11 (s) K12 (s) K13 (s) K21 (s) K22 (s) K23 (s) = 0. (12.28) K31 (s) K32 (s) K33 (s) It is worth noting that for simplicity reasons we will focus our study only on the speciﬁc case where α11 = α22 , α12 = α21 . (12.29) Having determined U (s), V (s), and W (s), the unique solution for u(x), v(x), and w(x) can be determined by using the inverse Laplace transform method. It is interesting to point out that the computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations. This will be used here to facilitate the computational work. For simplicity reasons we will focus our study only on the speciﬁc case where For the ﬁrst equation in (12.20): α11 = α13 , v(x) = 0, For the second equation in (12.20): α22 = α23 , u(x) = 0, (12.30) For the third equation in (12.20): α31 = α32 , w(x) = 0. Example 12.3 Solve the system of singular integral equations by using the Laplace transform method x 1 1 2√ x(15 + 8x2 ) = u(t) + w(t) dt, 1 1 15 (x − t) 2 (x − t) 2 0 x 1 9 4 1 3 dt, x (7 + 6x) = (12.31) 2 v(t) + 2 w(t) 28 (x − t) 3 (x − t) 3 0 x 4 1 1 1 x 4 (5 + 4x) = dt. 3 u(t) + 3 v(t) 5 (x − t) 4 (x − t) 4 0

372

12 Systems of Singular Integral Equations

Taking Laplace transform of both sides of each equation in (12.31) gives * √ −3 π 2 (U (s) + W (s)), πs 2 1 + 2 = s s 2π 2 2π 7 1 √ 2 s− 3 1 + = √ 2 s− 3 (V (s) + W (s)), (12.32) s 3Γ√ 3 √3Γ 3 1 2π 2π 5 1 3 s− 4 1 + = 3 s− 4 (U (s) + V (s)) s Γ 4 Γ 4 Solving this systems of equations for U (s), V (s) and W (s) by using any computer symbolic system, such as Maple or Mathematica, gives 1 1 2 U (s) = , V (s) = 2 , W (s) = 3 . (12.33) s s s By taking the inverse Laplace transform of both sides of each equation in (12.33), the exact solutions are given by (12.34) (u(x), v(x), w(x)) = (1, x, x2 ). Example 12.4 Solve the system of singular integral equations by using the Laplace transform method x 1 1 1 u(t) + w(t) dt, 3x 3 (2 + 3x) = 2 2 (x − t) 3 (x − t) 3 0 x 1 8 1 1 2 4 dt, x (15 − 32x ) = (12.35) 3 v(t) + 3 w(t) 15 (x − t) 4 (x − t) 4 0 x 1 10 3 1 u(t) + v(t) dt. x5 = 2 2 3 (x − t) 5 (x − t) 5 0 Taking Laplace transform of both sides of each equation in (12.35) gives 4π 2 2π 4 1 − √ 2 s 3 1 + s− 3 (U (s) + W (s)), = √ 2 s 3Γ 3 3Γ( 3 ) √ √ 2π − 1 2 2π − 5 6 (12.36) s 4 1 − 2 = s 4 (V (s) + W (s)), s Γ 34 Γ( 34 ) 8 3 2Γ 35 s− 5 = Γ 35 s− 5 (U (s) + V (s)) Solving this systems of equations for U (s), V (s), and W (s) by using any computer symbolic system, such as Maple or Mathematica, gives 2 2 2 1 6 1 6 1 6 U (s) = + 2 + 3 , V (s) = − 2 − 3 , W (s) = + 2 − 3 . (12.37) s s s s s s s s s By taking the inverse Laplace transform of both sides of each equation in (12.37), the exact solutions are given by (u(x), v(x), w(x)) = (1 + 2x + 3x2 , 1 − 2x − 3x2 , 1 + 2x − 3x2 ).

(12.38)

12.2 Systems of Generalized Abel Integral Equations

373

Exercises 12.2.2 Solve the system of singular integral equations by using the Laplace transform method x ⎧ 1 4 3 ⎪ 2 (24x2 + 35) = ⎪ x 1 (u(t) + w(t)) dt ⎪ 105 ⎪ ⎪ 0 (x − t) 2 ⎪ ⎪ x ⎪ ⎨ 27 8 1 x 3 (9x + 11) = 1. 1 (v(t) + w(t)) dt 440 ⎪ 0 (x − t) 3 ⎪ ⎪ ⎪ ⎪ x ⎪ 16 7 1 ⎪ ⎪ x 4 (8x + 11) = ⎩ 1 (u(t) + v(t)) dt 231 0 (x − t) 4 x ⎧ 1 4 1 ⎪ ⎪ x 2 (8x + 9) = 1 (u(t) + w(t)) dt ⎪ ⎪ 3 ⎪ 0 (x − t) 2 ⎪ ⎪ ⎪ x ⎨ 3 2 1 x 3 (27x + 35) = 2. 1 (v(t) + w(t)) dt 10 ⎪ 0 (x − t) 3 ⎪ ⎪ x ⎪ ⎪ ⎪ 1 4 3 ⎪ ⎪ ⎩ x 4 (4x + 5) = 1 (u(t) + v(t)) dt 3 0 (x − t) 4 x ⎧ 1 3 2 ⎪ ⎪ x 3 (81x3 + 99x2 + 132x + 220) = 1 (u(t) + w(t)) dt ⎪ ⎪ 440 ⎪ 0 (x − t) 3 ⎪ ⎪ ⎪ x ⎨ 16 7 1 x 4 (32x2 + 80x + 55) = 3. 1 (v(t) + w(t)) dt 1155 ⎪ 0 (x − t) 4 ⎪ ⎪ x ⎪ ⎪ ⎪ 1 5 4 ⎪ ⎪ x 5 (25x2 + 70x + 63) = ⎩ 1 (u(t) + v(t)) dt 252 0 (x − t) 5 x ⎧ 1 3 1 ⎪ ⎪ x 3 (81x3 + 90x2 + 105x + 140) = 2 (u(t) + w(t)) dt ⎪ ⎪ 140 ⎪ 0 (x − t) 3 ⎪ ⎪ ⎪ x ⎨ 16 5 1 x 4 (96x2 + 208x + 117) = 4. 3 (v(t) + w(t)) dt 585 ⎪ 0 (x − t) 4 ⎪ ⎪ x ⎪ ⎪ ⎪ 1 5 3 ⎪ ⎪ x 5 (25x2 + 65x + 52) = ⎩ 2 (u(t) + v(t)) dt 156 0 (x − t) 5 x ⎧ 1 4 1 ⎪ 2 (8x3 − 70x − 105) = ⎪ x − 1 (u(t) + w(t)) dt ⎪ ⎪ 35 ⎪ 0 (x − t) 2 ⎪ ⎪ x ⎪ ⎨ 1 9 1 x 3 (27x3 − 30x2 − 560) = 5. − 2 (v(t) + w(t)) dt 140 ⎪ 0 (x − t) 3 ⎪ ⎪ ⎪ ⎪ x ⎪ 1 8 3 ⎪ ⎪ x 4 (16x2 + 132x + 231) = ⎩ 1 (u(t) + v(t)) dt 231 0 (x − t) 4 x ⎧ 1 9 2 ⎪ 3 (3x2 − 8x − 20) = ⎪ x − 1 (u(t) + w(t)) dt ⎪ 40 ⎪ ⎪ 0 (x − t) 3 ⎪ ⎪ x ⎪ ⎨ 1 5 3 x 5 (50x2 + 65x − 208) = 6. − 2 (v(t) + w(t)) dt 312 ⎪ 0 (x − t) 5 ⎪ ⎪ ⎪ ⎪ x ⎪ 1 4 1 ⎪ ⎪ x 4 (64x2 − 36x + 135) = ⎩ 3 (u(t) + v(t)) dt 45 0 (x − t) 4

374 ⎧ 9 5 ⎪ ⎪ x 3 (81x2 ⎪ ⎪ 220 ⎪ ⎪ ⎪ ⎪ ⎨ 25 8 x 5 (25x2 7. 468 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 32 5 ⎪ ⎪ x 4 (96x2 ⎩ 585 ⎧ 27 5 ⎪ ⎪ x 3 (45x2 ⎪ ⎪ 440 ⎪ ⎪ ⎪ ⎪ ⎨ 25 8 x 5 (25x2 8. 468 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 32 5 ⎪ ⎪ x 4 (96x2 ⎩ 585

12 Systems of Singular Integral Equations

x

1 1 (u(t) + w(t)) dt (x − t) 3 x 1 + 30x + 39) = 2 (v(t) + w(t)) dt 0 (x − t) 5 x 1 + 208x + 117) = 3 (u(t) + v(t)) dt 0 (x − t) 4 x 1 + 33x + 44) = 1 (u(t) + v(t) + w(t)) dt 0 (x − t) 3 x 1 + 30x + 39) = 2 (v(t) + w(t)) dt 0 (x − t) 5 x 1 + 208x + 117) = 3 (u(t) + v(t)) dt 0 (x − t) 4 + 44) =

0

12.3 Systems of the Weakly Singular Volterra Integral Equations In this section we will study systems of the weakly singular integral equations in two unknowns u(x) and v(x). Generalization to any number of unknowns can be followed in a parallel manner. Recall that the kernel is called weakly singular as the singularity may be transformed away by a change of variable [2]. In this section we will use only the Laplace transform method and the Adomian decomposition method to handle this type of systems.

12.3.1 The Laplace Transform Method The system of weakly singular Volterra integral equations of the convolution type in two unknowns is of the form x (K11 (x, t)u(t) + K12 (x, t)v(t)) dt, u(x) = f1 (x) + 0 (12.39) x (K21 (x, t)u(t) + K22 (x, t)v(t)) dt. v(x) = f2 (x) + 0

The kernels Kij (x, t), 1 i, j 2 and the functions fi (x), i = 1, 2 are given real-valued functions. The kernels Kij are singular kernels given by 1 Kij = , 1 i, j 2. (12.40) (x − t)αij Taking Laplace transforms of both sides of the system (12.39) gives the linear system in U (s) and V (s) U (s) = F1 (s) + K11 (s)U (s) + K12 (s)V (s), (12.41) V (s) = F2 (s) + K21 (s)U (s) + K22 (s)V (s),

12.3 Systems of the Weakly Singular Volterra Integral Equations

375

or equivalently (1 − K11 (s))U (s) − K12 (s)V (s) = F1 (s), −K21 (s)U (s) + (1 − K22 (s))V (s) = F2 (s),

(12.42)

where U (s) = L{u(x)}, V (s) = L{v(x)}, Fi (s) = L{fi (x)}, 1 i 2, Kij (s) = L{Kij (x)}, 1 i, j 2.

(12.43)

Solving (12.41) for U (s) and V (s) by using Cramer’s rule gives F1 (s) F2 (s)

1

−K12 (s) 1 − K22 (s) , U (s) = 1 − K11 (s) −K (s) 12 −K21 (s) 1 − K22 (s) and

− K11 (s) F1 (s) −K21 (s) F2 (s) . V (s) = 1 − K11 (s) −K12 (s) −K21 (s) 1 − K22 (s)

(12.44)

(12.45)

Having determined U (s) and V (s), the unique solution for u(x) and v(x) can be determined by using the inverse Laplace transform method. It is interesting to point out that the computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations of this weakly singular equations. The linear system (12.41) has a unique solution for u(x) and v(x) if and only if 1 − K11 (s) −K12 (s) (12.46) −K21 (s) 1 − K22 (s) = 0. Example 12.5 Solve the system of weakly singular Volterra integral equations by using the Laplace transform method x 3 64 7 2 x 2 (4x + 3) + u(x) = x4 − u(t) + v(t) dt, 1 1 105 (x − t) 2 (x − t) 2 0 x 2 32 7 3 3 2 x (16x − 27) + dt. v(x) = x − 1 u(t) − 1 v(t) 315 (x − t) 2 (x − t) 2 0 (12.47) Taking Laplace transform of both sides of each equation in (12.47) gives * * √ 12 π 6 π π 4! U (s) − 2 V (s) = 5 − +1 , 1−3 9 s s s s s2 (12.48) * * √ 3! 6 π 8 π π U (s) + 1 + 3 V (s) = 4 − 9 −3 . −2 s s s s s2 Solving this system of equations for U (s) and V (s), by using any computer symbolic system, such as Maple or Mathematica, gives

376

12 Systems of Singular Integral Equations

4! 3! , V (s) = 4 . (12.49) s5 s By taking the inverse Laplace transform of both sides of each equation in (12.49), the exact solutions are given by (12.50) (u(x), v(x)) = (x4 , x3 ). U (s) =

Example 12.6 Solve the system of weakly singular Volterra integral equations by using the Laplace transform method 9 2 u(x) = 1 + 2x − x 3 (9x2 + 6x + 10) 20 x 1 2 + dt, 1 u(t) + 1 v(t) (x − t) 3 (x − t) 3 0 (12.51) 3 2 v(x) = 1 + 3x2 + x 3 (27x2 − 72x − 20) 40 x 2 1 dt. + 1 u(t) − 1 v(t) (x − t) 3 (x − t) 3 0 Taking Laplace transforms of both sides of each equation in (12.51) gives

2Γ 23 Γ 23 3Γ 23 1 3 4 1 U (s) − 1 + , + + 1− V (s) = − 2 2 5 s s2 s s2 s3 s3 s3

2Γ 23 Γ 23 Γ 23 1 6 6 6 V (s) = − 1 + . + − U (s) + 1 + − 2 2 5 s s3 s s2 s3 s3 s3 (12.52) Solving this system of equations for U (s) and V (s) we ﬁnd 3 6 1 1 U (s) = + 2 , V (s) = + 3 . (12.53) s s s s The inverse Laplace transform of (12.53) gives the exact solutions by (12.54) (u(x), v(x)) = (1 + 3x, 1 + 3x2 ). Example 12.7 Solve the system of weakly singular Volterra integral equations by using the Laplace transform method 25 8 6 1 u(x) = x + x2 − x 5 (130x 5 + 182x 5 − 210x + 273) 6552 x 1 1 dt, + 1 u(t) + 2 v(t) (x − t) 5 (x − t) 5 0 (12.55) 25 6 6 1 2 5 5 5 v(x) = x − x − x (55x + 66x − 140x + 154) 924 x 1 1 u(t) + v(t) dt. + 3 4 (x − t) 5 (x − t) 5 0

12.3 Systems of the Weakly Singular Volterra Integral Equations

377

Taking Laplace transforms of both sides of each equation in (12.55) and solving the resulting system of equations for U (s) and V (s) we obtain 1 2 1 2 U (s) = 2 + 3 , V (s) = 2 − 3 . (12.56) s s s s The inverse Laplace transform of (12.56) gives the exact solutions by (12.57) (u(x), v(x)) = (x + x2 , x − x2 ). Example 12.8 Use the Laplace transform method to solve the system 3 2 8 3 x 4 (32x2 − 44x + 77) u(x) = 1 + x − x2 + x 3 (9x2 − 12x − 20) − 40 231 x 2 1 dt, + 1 u(t) + 1 v(t) (x − t) 3 (x − t) 4 0 5 4 v(x) = 1 − x + x2 + x 5 (25x2 − 35x − 63) (12.58) 126 6 5 x 6 (72x2 − 102x + 187) + 935 x 2 1 dt. + 1 u(t) − 1 v(t) (x − t) 5 (x − t) 6 0 Taking Laplace transform of both sides of each equation in (12.58), proceeding as before, and solving the resulting system of equations for U (s) and V (s) we obtain 2 1 2 1 1 1 (12.59) U (s) = + 2 − 3 , V (s) = − 2 + 3 . s s s s s s The inverse Laplace transform of (12.59) gives the exact solutions by (u(x), v(x)) = (1 + x − x2 , 1 − x + x2 ).

(12.60)

Exercises 12.3.1 Solve the following systems of weakly singular Volterra integral equations by using the Laplace transform method ⎧ x 1 25 7 1 ⎪ 2 ⎪ 5 ⎪ u(x) = x + x − x + dt 3 u(t) + 3 v(t) ⎪ ⎨ 7 0 (x − t) 5 (x − t) 5 1. x ⎪ ⎪ 1 25 8 1 ⎪ ⎪ dt x5 + ⎩ v(x) = x − x2 − 2 u(t) + 2 v(t) 12 0 (x − t) 5 (x − t) 5 ⎧ x 1 4 3 1 ⎪ 2 ⎪ ⎪ u(x) = x − dt x 2 (4x + 5) + 1 u(t) + 1 v(t) ⎪ ⎨ 15 0 (x − t) 2 (x − t) 2 2. x ⎪ ⎪ 4 3 1 1 ⎪ ⎪ dt x 2 (4x − 5) + ⎩ v(x) = x − 1 u(t) − 1 v(t) 15 0 (x − t) 2 (x − t) 2

378 ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨ 3.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) = ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨

4.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) = ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨

5.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) = ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨

6.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) = ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨

7.

8.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) =

12 Systems of Singular Integral Equations x 1 5 4 1 2 1 + x − x5 + dt 1 u(t) + 1 v(t) 2 0 (x − t) 5 (x − t) 5 x 5 4 1 1 u(t) + v(t) dt 1 − x2 − x 5 + 1 1 2 0 (x − t) 5 (x − t) 5 √ x 2 x 1 2 (16x + 10x + 15) + dt 1 + 3x − 1 (u(t) + 2v(t)) 5 0 (x − t) 2 √ x 1 2 x (2u(t) − v(t)) dt 1 + 3x2 + (8x2 − 20x − 5) + 1 5 0 (x − t) 2 x √ 1 (u(t) + v(t)) dt 1 + x + x2 − 4 x + 1 0 (x − t) 2 x 1 8 3 x 2 (4x + 5) + 1 − x − x2 − (u(t) − v(t)) dt 1 15 0 (x − t) 2 x √ 1 (u(t) + 2v(t)) dt 1 + x + 2x2 + 2 x(2x − 3) + 1 0 (x − t) 2 √ x 2 x 1 2 2 dt (8x + 8x + 3) + 1 − 2x − x − 1 (2u(t) − v(t)) 3 0 (x − t) 2 x 9 2 1 (u(t) + 2v(t)) dt 1 + x − x2 − x 3 (3x2 − 4x + 20) + 1 40 0 (x − t) 3 x 1 3 2 2 2 3 x (27x − 36x − 20) + 1−x+x + dt 1 (2u(t) − v(t)) 40 0 (x − t) 3

⎧ 7 1 9 1 ⎪ x 3 (21x 3 + 280x 3 − 60x2 + 560) ⎪ u(x) = 6 + x2 − ⎪ ⎪ 280 ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ dt + 1 u(t) + 2 v(t) ⎪ ⎨ 0 (x − t) 3 (x − t) 3

11 1 5 1 ⎪ ⎪ ⎪ v(x) = 6 − x2 − x 5 (275x 5 + 2772x 5 − 700x2 + 5544) ⎪ ⎪ 924 ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ dt + ⎩ 3 u(t) + 4 v(t) 0 (x − t) 5 (x − t) 5

12.3.2 The Adomian Decomposition Method The Adomian decomposition method, as presented before, decomposes each solution as an inﬁnite sum of components, where these components are determined recurrently. This method can be used in its standard form, or combined with the noise terms phenomenon. It will be shown that the modiﬁed decomposition method is eﬀective and reliable in handling the systems of weakly singular Volterra integral equations. In view of this, the modiﬁed decomposition method will be used extensively in this section.

12.3 Systems of the Weakly Singular Volterra Integral Equations

379

We will focus our work on the system of the generalized weakly singular Volterra integral equations in two unknowns of the form x u(x) = f1 (x) + (K11 (x, t)u(t) + K12 (x, t)v(t)) dt, 0 (12.61) x (K21 (x, t)u(t) + K22 (x, t)v(t)) dt. v(x) = f2 (x) + 0

The kernels Kij (x, t), 1 i, j 2 and the functions fi (x), i = 1, 2 are given real-valued functions. The kernels Kij are singular kernels of the generalized form given by 1 Kij = 1 i, j 2. (12.62) α , [g(x) − g(t)] ij This type of systems arise in many mathematical physics and chemistry applications such as stereology, heat conduction, crystal growth and the radiation of heat from a semi-inﬁnite solid. The Adomian decomposition method [8] has been discussed extensively in this text. As will be seen later, the modiﬁed decomposition method is reliable and eﬀective in handling the system (12.61). In addition, the noise terms phenomenon will be used when noise terms appear. For revision purposes, we give a brief review of the modiﬁed decomposition method. In this method we usually split each of the source terms f1 (x) and f2 (x) into two parts fi1 (x) deﬁned by f1 (x) = f11 + f12 , f2 (x) = f21 + f22 . (12.63) Based on this, we use the modiﬁed recurrence relation as follows: u0 (x) = f11 (x),

v0 (x) = f21 (x), x (K11 (x, t)u0 (t) + K12 (x, t)v0 (t)) dt, u1 (x) = f12 (x) + v1 (x) = f22 (x) +

uk+1 (x) = vk+1 (x) =

0

x 0

(K21 (x, t)u0 (t) + K22 (x, t)v0 (t)) dt,

(12.64)

x

0

0

(K11 (x, t)uk (t) + K12 (x, t)vk (t)) dt,

k 1,

(K21 (x, t)uk (t) + K22 (x, t)vk (t)) dt,

k 1.

x

The following examples will illustrate the analysis presented above. Example 12.9 Use the modiﬁed method to solve the system of the generalized weakly singular integral equations

380

12 Systems of Singular Integral Equations

x 3 1 1 2 x x u(x) = e − (3e + 7)(e − 1) 3 + dt, 1 u(t) + 1 v(t) 10 (ex − et ) 3 (ex − et ) 3 0 x 1 9 1 5 v(x) = e2x + (ex − 1) 3 + u(t) − v(t) dt. 1 1 10 (ex − et ) 3 (ex − et ) 3 0 (12.65) We set the modiﬁed recurrence relation u0 (x) = ex , v0 (x) = e2x , x 3 1 1 2 x x 3 dt u1 (x) = − (3e + 7)(e − 1) + 1 u0 (t) + 1 v0 (t) 10 (ex − et ) 3 (ex − et ) 3 0 x

= 0, x 9 x 1 1 5 3 (e − 1) + v1 (x) = dt = 0. 1 u0 (t) − 1 v0 (t) 10 (ex − et ) 3 (ex − et ) 3 0 (12.66) The exact solutions are therefore given by (u(x), v(x)) = (ex , e2x ).

(12.67)

Example 12.10 Solve the system of the generalized weakly singular Volterra integral equations by using the modiﬁed decomposition method 2 3 3 2 u(x) = sin x + (cos x − 1) 3 − sin 3 x 2 2 x 1 1 + u(t) + v(t) dt, 1 1 (cos x − cos t) 3 (sin x − sin t) 3 0 (12.68) 1

1

v(x) = cos x + 3(cos x − 1) 3 − 3 sin 3 x x 1 1 + dt. 2 u(t) − 2 v(t) (cos x − cos t) 3 (sin x − sin t) 3 0 Proceeding as before, we set the modiﬁed recurrence relation u0 (x) = sin x, v0 (x) = cos x, 2 3 3 2 u1 (x) = (cos x − 1) 3 − sin 3 x 2 2 x 1 1 u (t) + v (t) dt = 0, + 0 0 1 1 (cos x − cos t) 3 (sin x − sin t) 3 0 1

1

v1 (x) = 3(cos x − 1) 3 − 3 sin 3 x x 1 1 dt = 0, + 2 u0 (t) − 2 v0 (t) (cos x − cos t) 3 (sin x − sin t) 3 0 uk+1 (x) = 0,

vk+1 (x) = 0,

k 1. (12.69)

The exact solutions are therefore given by (u(x), v(x)) = (sin x, cos x).

(12.70)

12.3 Systems of the Weakly Singular Volterra Integral Equations

381

Example 12.11 Solve the system of the generalized weakly singular Volterra integral equations by using the modiﬁed decomposition method x 5 4 5 u(x) = x4 − 2x 2 − x4 + u(t) + v(t) dt, 1 1 (x5 − t5 ) 2 (x6 − t6 ) 3 0 x 15 24 15 24 5 4 5 v(x) = x − 4x + 5x + dt. 1 u(t) − 1 v(t) (x5 − t5 ) 4 (x6 − t6 ) 5 0 (12.71) We next set the modiﬁed recurrence relation v0 (x) = x5 , u0 (x) = x4 , u1 (x) = 0, v1 (x) = 0, (12.72) uk+1 (x) = 0, vk+1 (x) = 0, k 1. The exact solutions are therefore given by (u(x), v(x)) = (x4 , x5 ).

(12.73)

Example 12.12 Solve the system of the generalized weakly singular Volterra integral equations by using the modiﬁed decomposition method 1

1

u(x) = ex − 2(ex − 1) 2 (1 + e− 2 x ) x 1 1 + dt, 1 u(t) + 1 v(t) (ex − et ) 2 (e−t − e−x ) 2 0 3 2 2 v(x) = e−x − (ex − 1) 3 (1 + e− 3 x ) 2 x 1 1 dt. + 1 u(t) + 1 v(t) (ex − et ) 3 (e−t − e−x ) 3 0 Now we use the modiﬁed recurrence relation u0 (x) = ex , v0 (x) = e−x , u1 (x) = 0, v1 (x) = 0, uk+1 (x) = 0, vk+1 (x) = 0, k 1. The exact solutions are therefore given by (u(x), v(x)) = (ex , e−x ).

(12.74)

(12.75)

(12.76)

Exercises 12.3.2 Solve the following systems of generalized weakly singular integral equations by using the modiﬁed decomposition method

382 ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨ 1.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) = ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨

2.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) = ⎧ ⎪ ⎪ ⎪ u(x) = ⎪ ⎨

3.

⎪ ⎪ ⎪ ⎪ ⎩ v(x) =

12 Systems of Singular Integral Equations x 1 π 2 2 2 1 + x − (x − 6) + dt 1 u(t) + 1 v(t) 4 0 (x2 − t2 ) 2 (x2 − t2 ) 2 x 9 2 1 4 u(t) − v(t) dt 1 − x2 − x 3 (3x2 + 4) + 1 1 8 0 (x − t) 3 (x − t) 3 x 1 2 1 dt x − 4x 2 (x + 1) + 1 u(t) + 1 v(t) 0 (x − t) 2 (x − t) 2 x 4 1 3 2 u(t) − v(t) dt 1+x− x 3 (9x − 5) + 1 1 10 0 (x − t) 3 (x − t) 3 x 1 4 4 3 u(t) + v(t) dt x− x 2 (16x + 5) + 1 1 15 0 (x − t) 2 (x − t) 2 x 1 9 5 1 x 3 (3x − 4) + x4 + u(t) − v(t) dt 1 1 40 0 (x − t) 3 (x − t) 3

⎧ 1 1 ⎪ u(x) = sin x + 2(cos x − 1) 2 − 2 sin 2 x ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ + dt, ⎪ 1 u(t) + 1 v(t) ⎪ ⎪ 0 (cos x − cos t) 2 (sin x − sin t) 2 ⎨ 4. 2 2 3 3 ⎪ v(x) = cos x + (cos x − 1) 3 + sin 3 x ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ + dt. ⎪ 1 u(t) − 1 v(t) ⎩ 0 (cos x − cos t) 3 (sin x − sin t) 3 ⎧ 2 1 3 ⎪ ⎪ u(x) = cos x + (cos x − 1) 3 − 2 sin 2 x ⎪ ⎪ 2 ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ + u(t) + v(t) dt, ⎪ 1 1 ⎪ ⎨ 0 (sin x − sin t) 2 (cos x − cos t) 3 5. 4 3 5 4 ⎪ ⎪ v(x) = sin x + (cos x − 1) 5 − sin 4 x ⎪ ⎪ ⎪ 4 3 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ dt. + 1 u(t) + 1 v(t) ⎩ 0 (sin x − sin t) 4 (cos x − cos t) 5 ⎧ 2 2 3 3 ⎪ ⎪ u(x) = sinh x − (cosh x − 1) 3 − sinh 3 x ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ + dt, ⎨ 1 u(t) + 1 v(t) 0 (cosh x − cosh t) 3 (sinh x − sinh t) 3 6. ⎪ 1 1 ⎪ ⎪ v(x) = cosh x − 3(cosh x − 1) 3 − 3 sinh 3 x ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ + dt. ⎪ 2 u(t) + 2 v(t) ⎩ 0 (cosh x − cosh t) 3 (sinh x − sinh t) 3

References

7.

383

⎧ 2 1 3 ⎪ u(x) = cosh x − (cosh x − 1) 3 − 2 sinh 2 x ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ + dt, 1 u(t) + 1 v(t) ⎪ ⎨ 0 (sinh x − sinh t) 2 (cosh x − cosh t) 3

4 3 4 5 ⎪ ⎪ ⎪ v(x) = sinh x − (cosh x − 1) 5 − sinh 4 x ⎪ ⎪ 4 3 ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ + dt. ⎩ 1 u(t) + 1 v(t) 0 (sinh x − sinh t) 4 (cosh x − cosh t) 5 ⎧ 2 3 −2x ⎪ ⎪ u(x) = ex − (ex − 1) 3 (1 + e 3 ) ⎪ ⎪ 2 ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ u(t) + v(t) dt, + 1 1 ⎪ ⎨ 0 (ex − et ) 3 (e−t − e−x ) 3 8. 3 3 4 ⎪ ⎪ v(x) = e−x − (ex − 1) 4 (1 − e− 4 x ) ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ 1 1 ⎪ ⎪ + u(t) − v(t) dt ⎩ 1 1 0 (ex − et ) 4 (e−t − e−x ) 4

References 1. E. Erdogan, Approximate solutions of singular integral equations, SIAM J. Appl. Math., 17 (1969) 1041–1059 2. R. Estrada and R. Kanwal, Singular Integral Equations, Birkhauser, Berlin, (2000). 3. A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 4. R.P. Kanwal, Linear Integral Equations, Birkhauser, Boston, (1997). 5. E.G. Ladopoulos, Singular Integra Equations, Springer, Berlin, (2000). 6. M. Masujima, Applied Mathematical Methods in Theoretical Physics, WileyVCH, Weinheim, (2005). 7. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).

Part II

Nonlinear Integral Equations

Chapter 13

Nonlinear Volterra Integral Equations

13.1 Introduction It is well known that linear and nonlinear Volterra integral equations arise in many scientiﬁc ﬁelds such as the population dynamics, spread of epidemics, and semi-conductor devices. Volterra started working on integral equations in 1884, but his serious study began in 1896. The name integral equation was given by du Bois-Reymond in 1888. However, the name Volterra integral equation was ﬁrst coined by Lalesco in 1908. The linear Volterra integral equations and the linear Volterra integrodiﬀerential equations were presented in Chapters 3 and 5 respectively. It is our goal in this chapter to study the nonlinear Volterra integral equations of the ﬁrst and the second kind. The nonlinear Volterra equations are characterized by at least one variable limit of integration. In the nonlinear Volterra integral equations of the second kind, the unknown function u(x) appears inside and outside the integral sign. The nonlinear Volterra integral equation of the second kind is represented by the form x K(x, t)F (u(t))dt. (13.1) u(x) = f (x) + 0

However, the nonlinear Volterra integral equations of the ﬁrst kind contains the nonlinear function F (u(x)) inside the integral sign. The nonlinear Volterra integral equation of the ﬁrst kind is expressed in the form x f (x) = K(x, t)F (u(t))dt. (13.2) 0

For these two kinds of equations, the kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x) such as u2 (x), sin(u(x)), and eu(x) .

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

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13 Nonlinear Volterra Integral Equations

13.2 Existence of the Solution for Nonlinear Volterra Integral Equations In this section we will present an existence theorem for the solution of nonlinear Volterra integral equations. The complete proof of this theorem can be found in [1–6]. However, in what follows, we present a brief summary of the conditions under which a solution exists for this equation. We ﬁrst rewrite the nonlinear Volterra integral equation of the second kind by x

u(x) = f (x) +

G(x, t, u(t))dt.

(13.3)

0

The speciﬁc conditions under which a solution exists for the nonlinear Volterra integral equation are: (i) The function f (x) is integrable and bounded in a x b. (ii) The function f (x) must satisfy the Lipschitz condition in the interval (a, b). This means that |f (x) − f (y)| < k|x − y|. (13.4) (iii) The function G(x, t, u(t)) is integrable and bounded |G(x, t, u(t))| < K in a x, t b. (iv) The function G(x, t, u(t)) must satisfy the Lipschitz condition (13.5) |G(x, t, z) − G(x, t, z )| < M |z − z |. The emphasis in this chapter will be on solving the nonlinear Volterra integral equations rather than proving theoretical concepts of convergence and existence. The theorems of uniqueness, existence, and convergence are important and can be found in the literature. The concern in this text will be on the determination of the solution u(x) of the nonlinear Volterra integral equation of the ﬁrst and the second kind.

13.3 Nonlinear Volterra Integral Equations of the Second Kind We begin our study on nonlinear Volterra integral equations of the second kindgiven by x K(x, t)F (u(t))dt, (13.6) u(x) = f (x) + 0

where the kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x) such as u3 (x), cos(u(x)), andeu(x) . The unknown function u(x), that will be determined, occurs inside and outside the integral sign.

13.3 Nonlinear Volterra Integral Equations of the Second Kind

389

The nonlinear Volterra equation (13.6) will be handled by using three distinct methods. The three methods are the successive approximations method, the series solution method, and the Adomian decomposition method (ADM). The latter will be combined with the modiﬁed decomposition method and the noise terms phenomenon.

13.3.1 The Successive Approximations Method The successive approximations method [7], or the Picard iteration method was used before in Chapters 3 and 4. This method solves any problem by ﬁnding successive approximations to the solution by starting with an initial guess, called the zeroth approximation. As will be seen later, the zeroth approximation is any selective real-valued function that will be used in a recurrence relation to determine the other approximations. Given the nonlinear Volterra integral equation of the second kind x u(x) = f (x) + K(x, t)F (u(t))dt, (13.7) 0

where u(x) is the unknown function to be determined and K(x, t) is the kernel. The successive approximations method introduces the recurrence relation x K(x, t)F (un (t))dt, n 0, (13.8) un+1 (x) = f (x) + 0

where the zeroth approximation u0 (x) can be any selective real valued function. We always start with an initial guess for u0 (x), mostly we select 0, 1, or x for u0 (x). Using this selection of u0 (x) into (13.8), several successive approximations uk , k 1 will be determined as x u1 (x) = f (x) + K(x, t)F (u0 (t))dt, 0

u2 (x) = f (x) +

.. .

x

K(x, t)F (u2 (t))dt,

0

un+1 (x) = f (x) +

K(x, t)F (u1 (t))dt,

0

u3 (x) = f (x) +

x

(13.9)

x

0

K(x, t)F (un (t))dt.

Consequently, the solution u(x) is obtained by using u(x) = lim un+1 (x). n→∞

(13.10)

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13 Nonlinear Volterra Integral Equations

The question of convergence of un+1 (x) was examined in Chapter 3. The successive approximations method, or the Picard iteration method will be illustrated by the following examples. Example 13.1 Use the successive approximations method to solve the nonlinear Volterra integral equation x 1 x 3x u(x) = e + x(1 − e ) + xu3 (t)dt. (13.11) 3 0 For the zeroth approximation u0 (x), we can select (13.12) u0 (x) = 1. The method of successive approximations admits the use of the iteration formula x 1 x 3x xu3n (t)dt, n 0. (13.13) un+1 (x) = e + x(1 − e ) + 3 0 Substituting (13.12) into (13.13) we obtain the approximations u0 (x) = 1, x 1 x 3x u1 (x) = e + x(1 − e ) + xu30 (t)dt 3 0 4 35 67 1 = 1 + x + x2 − x3 − x4 − x5 + · · · , 2! 3 24 60 x 1 u2 (x) = ex + x(1 − e3x ) + xu31 (t)dt (13.14) 3 0 1 1 1 67 = 1 + x + x2 + x3 + x4 − x5 + · · · , 2! 3! 4! 60 x 1 u3 (x) = ex + x(1 − e3x ) + xu32 (t)dt 3 0 1 1 1 1 1 = 1 + x + x2 + x3 + x4 + x5 + x6 + · · · , 2! 3! 4! 5! 6! and so on. Consequently, the solution u(x) of (13.11) is given by u(x) = lim un (x) = ex . n→∞

(13.15)

Example 13.2 Use the successive approximations method to solve the nonlinear Volterra integral equation x 16 4 u(x) = 4x − x3 − x4 + (x − t + 1)u2 (t)dt. (13.16) 3 3 0 For the zeroth approximation u0 (x), we can select u0 (x) = 0.

(13.17)

13.3 Nonlinear Volterra Integral Equations of the Second Kind

391

The method of successive approximations admits the use of the iteration formula x 16 4 un+1 (x) = 4x − x3 − x4 + (x − t + 1)u2n (t)dt, n 0. (13.18) 3 3 0 Substituting (13.17) into (13.18) we obtain the approximations u0 (x) = 0, x 16 3 4 4 u1 (x) = 4x − x − x + (x − t + 1)u20 (t)dt 3 3 0 4 16 = 4x − x3 − x4 , 3 3 x 16 3 4 4 u2 (x) = 4x − x − x + (x − t + 1)u21 (t)dt 3 3 0 4 4 4 4 128 5 16 6 16 3 16 3 x − x + x − x − x − x + ··· , = 4x + 3 3 3 3 15 5 x 16 3 4 4 (x − t + 1)u22 (t)dt u3 (x) = 4x − x − x + 3 3 0 128 5 128 5 16 6 16 6 x − x + x − x + ··· . = 4x + 15 15 5 5 (13.19) By canceling the noise terms, the solution u(x) of (13.16) is given by (13.20) u(x) = lim un (x) = 4x. n→∞

Example 13.3 Use the successive approximations method to solve the nonlinear Volterra integral equation x 1 1 1 (x − t)u2 (t)dt. (13.21) u(x) = cos(x) + cos(2x) − x2 − + 8 4 8 0 For the zeroth approximation u0 (x), we can select (13.22) u0 (x) = 1. The method of successive approximations admits the use of the iteration formula x 1 1 1 (x − t)u2n (t)dt, n 0. (13.23) un+1 (x) = cos(x) + cos(2x) − x2 − + 8 4 8 0 Substituting (13.22) into (13.23) we obtain the approximations u0 (x) = 1, 1 1 1 u1 (x) = 1 − x2 + x4 − x6 + · · · , 2! 8 80 (13.24) 1 2 1 4 1 6 u2 (x) = 1 − x + x + x + ··· , 2! 4! 240 1 2 1 4 1 u3 (x) = 1 − x + x − x6 + · · · . 2! 4! 6!

392

13 Nonlinear Volterra Integral Equations

Consequently, the solution u(x) of (13.21) is given by u(x) = lim un (x) = cos x. n→∞

(13.25)

Example 13.4 Use the successive approximations method to solve the nonlinear Volterra integral equation x 1 2 1 4 1 6 u(x) = 1 + x − x − x + (x − t)u2 (t)dt. (13.26) 2 6 30 0 For the zeroth approximation u0 (x), we can select u0 (x) = 1.

(13.27)

The method of successive approximations admits the use of the iteration formula x 1 1 1 (x − t)u2n (t)dt, n 0. (13.28) un+1 (x) = 1 + x2 − x4 − x6 + 2 6 30 0 Substituting (13.27) into (13.28) we obtain the approximations u0 (x) = 1, 1 1 u1 (x) = 1 + x2 − x4 − x6 , 6 30 1 4 1 4 1 (13.29) u2 (x) = 1 + x2 + x − x − x6 + · · · , 6 6 90 1 4 1 4 1 6 1 6 2 u3 (x) = 1 + x + x − x + x − x + ··· , 6 6 90 90 and so on. By canceling the noise terms, the solution u(x) of (13.26) is given by (13.30) u(x) = lim un (x) = 1 + x2 . n→∞

Exercises 13.3.1 Solve the following nonlinear Volterra integral equations by using the successive approximations method x 1. u(x) = 1 + u2 (t)dt, |x| < 1 0

2. u(x) = 1 + 3x −

3 1 2 x − x3 − x4 + 2 4

3. u(x) = 1 + 2x −

7 2 3 x − 4x3 − x4 + 2 4

x 0

(x − t)u2 (t)dt

x

0 x

(x − t + 1)u2 (t)dt

1 3 1 5 (x − t − 1)u3 (t)dt x − x + 2 20 0 x 1 1 (x − t)u2 (t)dt 5. u(x) = sin x + sin2 x − x2 + 4 4 0 4. u(x) = 1 + 2x + x2 +

13.3 Nonlinear Volterra Integral Equations of the Second Kind x 1 1 1 6. u(x) = sin x + cos x + sin 2x − x − x2 + (x − t)u2 (t)dt 4 2 2 0 x 1 1 1 7. u(x) = cos x − sin x − sin 2x + x − x2 + (x − t)u2 (t)dt 4 2 2 0 x 3 1 5 8. u(x) = 3 cos x + cos2 x − − x2 + (x − t)u2 (t)dt 4 4 4 0 x 5 1 1 (x − t)u2 (t)dt 9. u(x) = sinh x − cosh2 x + − 2x − x2 + 4 4 4 0 x 1 1 1 10. u(x) = ex − e2x + + x + (x − t)u2 (t)dt 4 4 2 0 x 1 1 1 11. u(x) = ex − e3x + + x + (x − t)u3 (t)dt 9 9 3 0 x 7 1 1 3 12. u(x) = ex + e2x + + x − x2 + (x − t − 1)u2 (t)dt 4 4 2 2 0

393

13.3.2 The Series Solution Method The series solution method was applied in Chapters 3, 4 and 5 to handle linear Volterra and Fredholm integral equations [1–2]. In this section, the series solution method will be applied in a similar manner to handle the nonlinear Volterra integral equations. Recall that the generic form of Taylor series at x = 0 can be written as ∞ u(x) = a n xn . (13.31) n=0

We will assume that the solution u(x) of the nonlinear Volterra integral equation x K(x, t)F (u(t))dt,

u(x) = f (x) +

(13.32)

0

is analytic, and therefore possesses a Taylor series of the form given in (13.31), where the coeﬃcients an will be determined recurrently. Substituting (13.31) into both sides of (13.32) gives ∞ x ∞ n n an x = T (f (x)) + K(x, t) F an t dt, (13.33) 0

n=0

or for simplicity we use 2

a0 + a1 x + a2 x + · · · = T (f (x)) +

n=0

0

x

K(x, t) F (a0 + a1 t + a2 t2 + · · · ) dt,

(13.34) where T (f (x)) is the Taylor series for f (x). The integral equation (13.32) will be converted to a traditional integral in (13.33) or (13.34) where instead of integrating the nonlinear term F (u(x)), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x)

394

13 Nonlinear Volterra Integral Equations

includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integral in (13.33) or (13.34), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x in both sides of the resulting equation to obtain a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (13.31). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we determine, the higher accuracy level we achieve. Example 13.5 Solve the following nonlinear Volterra integral equation by using the series solution method x 1 1 1 (x − t)u2 (t)dt. (13.35) u(x) = 1 + x − x2 − x3 − x4 + 2 3 12 0 Using the series form (13.31) into both sides of (13.35) gives 1 1 1 a0 + a1 x + a2 x2 + · · · = 1 + x − x2 − x3 − x4 2 3 12 x

2 + (x − t) a0 + a1 t + a2 t2 + a3 t3 + · · · dt, 0

(13.36) where by integrating the integral at the right side, and collecting like powers of x we obtain a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · 1 1 1 = 1 + x + (a20 − 1)x2 + (a0 a1 − 1)x3 + (a21 + 2a0 a2 − 1)x4 + · · · . 2 3 12 (13.37) Equating the coeﬃcients of like powers of x in both sides yields a0 = 1, a1 = 1, The exact solution is given by

an = 0,

u(x) = 1 + x.

for n 2.

(13.38) (13.39)

Example 13.6 We next consider the nonlinear Volterra integral equation x 1 3x 1 1 x (x − t)u3 (t)dt. (13.40) u(x) = + x + e − e + 9 3 9 0 Substituting the series (13.31) into both sides of (13.40) noting that u3 (x) = a0 3 +3a0 2 a1 x+(3a0 a1 2 +3a0 2 a2 )x2 +(a31 +6a1 a2 a0 +3a3 a20 )x3 +· · · , (13.41)

13.3 Nonlinear Volterra Integral Equations of the Second Kind

395

gives 1 1 1 a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · = + x + ex − e3x 9 3 9 x

3 + (x − t) a0 + a1 t + a2 t2 + a3 t3 + · · · dt.

(13.42)

0

Integrating the integral at the right hand side of (13.42), using the Taylor series of ex and e3x , and equating the coeﬃcients of like powers of x we ﬁnd a0 = 1, a1 = 1, 1 1 a2 = a20 = , 2! 2! 1 1 1 a3 = − + a20 a1 = , 3 2 3! (13.43) 1 1 1 2 a4 = − + a0 (a1 + a0 a2 ) = , 3 4 4! .. . 1 n! This gives the solution in a series form 1 1 1 u(x) = 1 + x + x2 + x3 + x4 + · · · . 2! 3! 4! Consequently, the exact solution is given by an =

u(x) = ex .

(13.44) (13.45)

Example 13.7 We next consider the nonlinear Volterra integral equation x 1 1 2 (x − t)u2 (t)dt. (13.46) u(x) = (x − x ) + cos x − sin x − sin(2x) + 2 4 0 Substituting the series (13.31) into both sides of (13.46) gives 1 a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · = (x − x2 ) + cos x − sin x 2 x (13.47)

2 1 (x − t) a0 + a1 t + a2 t2 + a3 t3 + · · · dt. − sin 2x + 4 0 Integrating the integral at the right hand side of (13.47), using the Taylor series of sin x, cos x and sin 2x, and equating the coeﬃcients of like powers of x we ﬁnd a0 = 1, a1 = −1, 1 1 a2 = −1 + a20 = − , 2! 2! 1 1 1 a 3 = + a0 a 1 = , (13.48) 2 3 3!

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13 Nonlinear Volterra Integral Equations

1 1 1 1 + a0 a2 + a21 = , 4! 6 12 4! 1 3 1 1 a5 = − + a1 a2 + a0 a3 = − , 40 10 10 5! and so on. The solution in a series form is given by 1 1 1 1 u(x) = 1 − x2 + x4 + · · · − x − x3 + x5 + · · · , 2! 4! 3! 5! that converges to the exact solution u(x) = cos x − sin x. a4 =

(13.49)

(13.50)

Example 13.8 We ﬁnally consider the nonlinear Volterra integral equation x 1 9 1 2 (x − t)u2 (t)dt. (13.51) u(x) = + 2x − x − sinh x − cosh 2x + 8 4 8 0 Substituting the series (13.31) into both sides of (13.51) gives 9 1 a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · = + 2x − x2 − sinh x 8 4 x (13.52)

2 1 − cosh 2x + (x − t) a0 + a1 t + a2 t2 + a3 t3 + · · · dt. 8 0 Integrating the integral at the right hand side of (13.52), using the Taylor series of sinh x and cosh 2x, and equating the coeﬃcients of like powers of x we ﬁnd a0 = 1, a1 = 1, 1 1 a2 = − + a20 = 0, 2 2! 1 1 1 a 3 = − + a0 a 1 = , (13.53) 6 3 3! 1 1 1 a4 = − + a0 a2 + a21 = 0, 12 6 12 1 1 1 1 a5 = − + a1 a2 + a0 a3 = , 120 10 10 5! and so on. The solution in a series form is given by 1 5 1 3 (13.54) u(x) = 1 + x + x + x + · · · , 3! 5! that converges to the exact solution u(x) = 1 + sinh x. (13.55)

Exercises 13.3.2 Solve the following nonlinear Volterra integral equations by using the series solution method

13.3 Nonlinear Volterra Integral Equations of the Second Kind x 1 1 1 4 1. u(x) = 1 + x − x2 − x3 − (x − t)u2 (t)dt x + 2 3 12 0 x 1 1 1 6 x + 2. u(x) = 1 + x2 − x4 − (x − t)u2 (t)dt 2 6 30 0 x 7 3 3. u(x) = 1 + 2x − x2 − 4x3 − x4 + (x − t + 1)u2 (t)dt 2 4 0 x 2 1 (x − t)u3 (t)dt 4. u(x) = 1 + 2x − x2 − x3 − x4 − x5 + 2 5 0 x 1 1 5. u(x) = sin x + sin2 x − x2 + (x − t)u2 (t)dt 4 4 0 x 1 1 1 6. u(x) = sin x + cos x + sin 2x − x − x2 + (x − t)u2 (t)dt 4 2 2 0 x 1 1 1 7. u(x) = − x2 + cosh x − cosh 2x + (x − t)u2 (t)dt 8 4 8 0 x 1 1 1 8. u(x) = x − x3 + cosh x − sinh 2x + (x − t)2 u2 (t)dt 4 6 8 0 x 5 1 1 (x − t)u2 (t)dt 9. u(x) = sinh x − cosh2 x + − 2x − x2 + 4 4 4 0 x 1 1 1 1 10. u(x) = + x + x2 + ex − e2x + (x − t)2 u2 (t)dt 4 2 2 4 0 x 1 1 1 11. u(x) = ex − e3x + + x + (x − t)u3 (t)dt 9 9 3 0 x 1 1 1 12. u(x) = − x + e−x − e−2x + (x − t)u2 (t)dt 4 2 4 0

397

13.3.3 The Adomian Decomposition Method The Adomian decomposition method has been outlined before in previous chapters and has been applied to a wide class of linear Volterra and Fredholm integral equations. The method usually decomposes the unknown function u(x) into an inﬁnite sum of components that will be determined recursively through iterations as discussed before. The Adomian decomposition method will be applied in this chapter and in the coming chapters to handle nonlinear integral equations. Although the linear term u(x) is represented by an inﬁnite sum of components, the nonlinear terms such as u2 , u3 , u4 , sin u, eu , etc. that appear in the equation, should be expressed by a special representation, called the Adomian polynomials An , n 0. Adomian introduced a formal algorithm to establish a reliable representation for all forms of nonlinear terms. Other techniques to evaluate Adomian polynomials were developed, but Adomian technique remains the commonly used one. In this text, we will use the Adomian algorithm to evaluate Adomian polynomials. The representation of the nonlinear

398

13 Nonlinear Volterra Integral Equations

terms by Adomian polynomials is necessary to handle the nonlinear integral equations in a reliable way. In the following, the Adomian algorithm for calculating the so-called Adomian polynomials for representing nonlinear terms will be introduced in details. The algorithm will be explained by illustrative examples that will cover a wide variety of nonlinear forms.

Calculation of Adomian Polynomials The Adomian decomposition method [8] assumes that the unknown linear function u(x) may be represented by the inﬁnite decomposition series ∞ u(x) = un (x), (13.56) n=0

where the components un (x), n 0 will be computed in a recursive way. However, the nonlinear term F (u(x)), such as u2 , u3 , u4 , sin u, eu, etc. can be expressed by an inﬁnite series of the so-called Adomian polynomials An given by n 1 dn i F An = λ ui , n = 0, 1, 2, · · · , (13.57) n! dλn i=0 λ=0

where the so-called Adomian polynomials An can be evaluated for all forms of nonlinearity. The general formula (13.57) can be easily used as follows. Assuming that the nonlinear function is F (u(x)), therefore by using (13.57), Adomian polynomials are given by A0 = F (u0 ), A1 = u1 F (u0 ), 1 A2 = u2 F (u0 ) + u21 F (u0 ), 2! 1 (13.58) A3 = u3 F (u0 ) + u1 u2 F (u0 ) + u31 F (u0 ), 3! 1 2 1 A4 = u4 F (u0 ) + u2 + u1 u3 F (u0 ) + u21 u2 F (u0 ) 2! 2! 1 + u41 F (iv) (u0 ). 4! Two important observations can be made here. First, A0 depends only on u0 , A1 depends only on u0 and u1 , A2 depends only on u0 , u1 , and u2 , and so on. Second, substituting (13.58) into (13.57) gives F (u) = A0 + A1 + A2 + A3 + · · · = F (u0 ) + (u1 + u2 + u3 + · · · )F (u0 ) 1 + (u21 + 2u1 u2 + 2u1 u3 + u22 + · · · )F (u0 ) + · · · 2!

13.3 Nonlinear Volterra Integral Equations of the Second Kind

399

1 3 (u + 3u21 u2 + 3u21 u3 + 6u1 u2 u3 + · · · )F (u0 ) + · · · 3! 1 1 (13.59) = F (u0 ) + (u − u0 )F (u0 ) + (u − u0 )2 F (u0 ) + · · · . 2! The last expansion conﬁrms a fact that the series in An polynomials is a Taylor series about a function u0 and not about a point as in the standard Taylor series. The few Adomian polynomials given above in (13.58) clearly show that the sum of the subscripts of the components of u(x) of each term of An is equal to n. In the following, we will calculate Adomian polynomials for several nonlinear terms that may arise in nonlinear integral equations. +

Case 1. The ﬁrst four Adomian polynomials for F (u) = u2 are given by A0 = F (u0 ) = u20 , A1 = u1 F (u0 ) = 2u0 u1 , 1 A2 = u2 F (u0 ) + u21 F (u0 ) = 2u0 u2 + u21 , 2! 1 A3 = u3 F (u0 ) + u1 u2 F (u0 ) + u31 F (u0 ) = 2u0 u3 + 2u1 u2 . 3! Case 2. The ﬁrst four Adomian polynomials for F (u) = u3 are given by A0 = F (u0 ) = u30 , A1 = u1 F (u0 ) = 3u20 u1 , 1 A2 = u2 F (u0 ) + u21 F (u0 ) = 3u20 u2 + 3u0 u21 , 2! 1 A3 = u3 F (u0 ) + u1 u2 F (u0 ) + u31 F (u0 ) = 3u20 u3 + 6u0 u1 u2 + u31 , 3! 1 1 1 A4 = u4 F (u0 ) + u2 + u1 u3 F (u0 ) + u21 u2 F (u0 ) + u41 F (iv) (u0 ). 2! 2 2! 4! Case 3. The ﬁrst four Adomian polynomials for F (u) = u4 are given by A0 A1 A2 A3

= u40 , = 4u30 u1 , = 4u30 u2 + 6u20 u21 , = 4u30 u3 + 4u31 u0 + 12u20 u1 u2 .

Case 4. The ﬁrst four Adomian polynomials for F (u) = sin u are given by A0 = sin u0 ,

400

13 Nonlinear Volterra Integral Equations

A1 = u1 cos u0 , A2 = u2 cos u0 −

1 2 u sin u0 , 2! 1

A3 = u3 cos u0 − u1 u2 sin u0 −

1 3 u cos u0 . 3! 1

Case 5. The ﬁrst four Adomian polynomials for F (u) = cos u are given by A0 = cos u0 , A1 = −u1 sin u0 , A2 = −u2 sin u0 −

1 2 u cos u0 , 2! 1

A3 = −u3 sin u0 − u1 u2 cos u0 +

1 3 u sin u0 . 3! 1

Case 6. The ﬁrst four Adomian polynomials for F (u) = eu are given by A0 = eu0 , A1 = u1 eu0 , 1 A2 = u2 + u21 eu0 , 2! 1 A3 = u3 + u1 u2 + u31 eu0 . 3!

Applying the Adomian Decomposition Method In what follows we present an outline for using the Adomian decomposition method for solving the nonlinear Volterra integral equation x K(x, t) F (u(t))dt, (13.60) u(x) = f (x) + 0

where F (u(t)) is a nonlinear function of u(x). The nonlinear Volterra integral equation (13.60) contains the linear term u(x) and the nonlinear function F (u(x)). The linear term u(x) of (13.60) can be represented normally by the decomposition series ∞ u(x) = un (x), (13.61) n=0

where the components un (x), n 0 can be easily computed in a recursive manner as discussed before. However, the nonlinear function F (u(x)) of (13.60) should be represented by the so-called Adomian polynomials An by using the algorithm

13.3 Nonlinear Volterra Integral Equations of the Second Kind

An =

1 dn n! dλn

F

n i=0

401

, n = 0, 1, 2, · · · ,

λi ui

(13.62)

λ=0

that was given above in (13.57). The standard Adomian method will be used for the ﬁrst example. However, for the next three examples, we will use the modiﬁed decomposition method and the noise terms phenomenon to minimize the use of Adomian polynomials. Example 13.9 Use the Adomian decomposition method to solve the nonlinear Volterra integral equation x u(x) = x + u2 (t)dt. (13.63) 0

Substituting the series (13.61) and the Adomian polynomials (13.62) into the left side and the right side of (13.63) respectively gives x ∞ ∞ un (x) = x + An (t)dt, (13.64) n=0

0

n=0 2

where An are the Adomian polynomials for u (x) as shown above. Using the Adomian decomposition method we set x u0 (x) = x, uk+1 (x) = Ak (t)dt, k 0. (13.65) 0

This in turn gives u0 (x) = x, x 1 u20 (t)dt = x3 , u1 (x) = 3 0 x 2 5 (2u0 (t)u1 (t))dt = u2 (x) = x , 15 0 x 17 7 u3 (x) = x , (2u0 (t)u2 (t) + u21 (t))dt = 315 0 and so on. The solution in a series form is given by 2 17 7 1 x + ··· , u(x) = x + x3 + x5 + 3 15 315 that converges to the exact solution u(x) = tan x.

(13.66)

(13.67) (13.68)

Example 13.10 We consider the nonlinear Volterra integral equation x 3 1 (x − t)u2 (t)dt. (13.69) u(x) = 1 + 3x − x2 − x3 − x4 + 2 4 0 Substituting the series (13.61) and the Adomian polynomials (13.62) into the left side and the right side of (13.69) respectively gives

402

13 Nonlinear Volterra Integral Equations ∞

1 3 un (x) = 1 + 3x − x2 − x3 − x4 + 2 4 n=0

0

x

(x − t)

∞

An (t)dt,

(13.70)

n=0

where An are the Adomian polynomials for u2 (x) as shown above. Using the modiﬁed decomposition method we set 1 u0 (x) = 1 + 3x − x2 , 2 x 3 u1 (x) = −x3 − x4 + (x − t)A0 (t)dt 4 0 x 3 1 5 3 1 6 (x − t)u20 (t)dt = x2 − x4 − x5 + = −x3 − x4 + x , 4 2 6 20 120 0 x uk+1 (x) = (x − t)Ak (t)dt, k 1. 0

(13.71) We can observe the appearance of the noise terms − 12 x2 and 12 x2 between u0 (x) and u1 (x). By canceling the noise term − 12 x2 from u0 (x), we can show that u(x) = 1 + 3x, (13.72) is the exact solution that satisﬁes the integral equation. It is worth noting that we did not use the Adomian polynomials. This is due to the fact the modiﬁed decomposition method and the noise terms phenomenon accelerate the convergence of the solution. Example 13.11 We consider the nonlinear Volterra integral equation x 1 1 (x − t)u2 (t)dt. (13.73) u(x) = sin x + sin2 x − x2 + 4 4 0 Substituting the series (13.61) and the Adomian polynomials (13.62) into the left side and the right side of (13.73) respectively we ﬁnd x ∞ ∞ 1 1 un (x) = sin x + sin2 x − x2 + (x − t) An (t)dt, (13.74) 4 4 0 n=0 n=0 where An are the Adomian polynomials for u2 (x) as shown above. Using the modiﬁed decomposition method we set 1 u0 (x) = sin x + sin2 x, 4 x 1 2 (x − t)A0 (t)dt u1 (x) = − x + 4 0 x (13.75) 1 2 1 =− x + (x − t)u20 (t)dt = − sin2 x + · · · , 4 4 0 x (x − t)Ak (t)dt, k 1. uk+1 (x) = 0

13.3 Nonlinear Volterra Integral Equations of the Second Kind

403

We can observe the appearance of the noise terms 14 sin2 x and − 41 sin2 x between u0 (x) and u1 (x) respectively. By canceling the noise term 14 sin2 x from u0 (x), we can show that u(x) = sin x, is the exact solution that satisﬁes the integral equation.

(13.76)

Example 13.12 We consider the nonlinear Volterra integral equation x 1 (x − t)u2 (t)dt. (13.77) u(x) = sec x − tan x − x2 + 2 0 Substituting the series (13.61) and the Adomian polynomials (13.62) into the left side and the right side of (13.77) respectively gives x ∞ ∞ 1 un (x) = sec x − tan x − x2 + (x − t) An (t)dt, (13.78) 2 0 n=0 n=0 where An are the Adomian polynomials for u2 (x) as shown above. Using the modiﬁed decomposition method we set u0 (x) = sec x, x 1 u1 (x) = − tan x − x2 + (x − t)A0 (t)dt 4 (13.79) 0 x 1 (x − t)u20 (t)dt = 0. = − tan x − x2 + 2 0 The other components uk , k 2 vanish as a result. Consequently, the exact solution is given by u(x) = sec x. (13.80)

Exercises 13.3.3 Solve the following nonlinear Volterra integral equations of the second kind by using the Adomian decomposition method, the modiﬁed Adomian decomposition method, or by using the noise terms phenomenon x 1. u(x) = x − u2 (t)dt 0

1 1 6 1 x + 2. u(x) = 1 + x2 − x4 − 2 6 30 3. u(x) = 1 + x2 +

x 0

(x − t)u2 (t)dt

1 4 1 6 1 8 x + x + x + 6 10 42

1 2 2 x − x3 − x4 − x5 + 2 5 x ex−t eu(t) dt 5. u(x) = x(1 − ex ) +

4. u(x) = 1 + 2x −

0

x

(xt2 − x2 t)u2 (t)dt

0

x

0

(x − t)u3 (t)dt

404

13 Nonlinear Volterra Integral Equations x 1 1 1 6. u(x) = sin x + cos x + sin 2x − x − x2 + (x − t)u2 (t)dt 4 2 2 0 x 2 2 2 7. u(x) = 1 + x2 − xex + ex −t −1 eu(t) dt 8. u(x) = ex − xex +

0

x

0

ex−2t u2 (t)dt

x 5 1 1 (x − t)u2 (t)dt cosh2 x + − 2x − x2 + 4 4 4 0 x 2 1 1 10. u(x) = sin x + cos x − cos2 x − + sin(x − t)u2 (t)dt 3 3 3 0 x 1 1 3 2 11. u(x) = sin x − cos2 x + sin 2x − ex + + ex−t u(t)dt 5 5 5 5 0 x 3 1 12. u(x) = cos x − sin 2x − sinh x + cosh(x − t)u2 (t)dt 5 5 0 9. u(x) = sinh x −

13.4 Nonlinear Volterra Integral Equations of the First Kind The standard form of the nonlinear Volterra integral equation of the ﬁrst kind is given by x K(x, t)F (u(t))dt, (13.81) f (x) = 0

where the kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x). Recall that the unknown function u(x) occurs only inside the integral sign for the Volterra integral equation of the ﬁrst kind. The linear Volterra integral equation of the ﬁrst kind is presented in Section 3.3 where three main methods were used for handling this kind of equations. To determine a solution for the nonlinear Volterra integral equation of the ﬁrst kind (13.81), we ﬁrst convert it to a linear Volterra integral equation of the ﬁrst kind of the form x

f (x) =

K(x, t)v(t)dt,

(13.82)

0

by using the transformation This in turn means that

v(x) = F (u(x)).

(13.83)

u(x) = F −1 (v(x)).

(13.84)

It is worth noting that the Volterra integral equation of the ﬁrst kind (13.82) can be solved by any method that was studied in Section 3.3. However, in this section we will handle Eq. (13.82) by the Laplace transform method and the

13.4 Nonlinear Volterra Integral Equations of the First Kind

405

conversion to Volterra integral equation of the second kind. Other methods can be found in the literature and in this text as well.

13.4.1 The Laplace Transform Method The Laplace transform method is a powerful technique that we used before for solving Volterra integral equations of the ﬁrst and the second kinds. We assume that the kernel K(x, t) is a diﬀerence kernel. Taking the Laplace transforms of both sides of (13.82) gives L{f (x)} = L{K(x − t)} × L{v(x)}, (13.85) so that V (s) =

F (s) , K(s)

(13.86)

where F (s) = L{f (x)}, K(s) = L{K(x)}, V (s) = L{v(x)}.

(13.87)

Taking the inverse Laplace transform of both sides of (13.86) gives v(x). The solution u(x) is obtained by using (13.84). It is obvious that the Laplace transform method works eﬀectively provided that F (s) = 0. (13.88) lim s→∞ K(s) The Laplace transform method will be used for studying the following nonlinear Volterra integral equations of the ﬁrst kind. Example 13.13 Solve the nonlinear Volterra integral equation of the ﬁrst kind by using the Laplace transform method x 1 2x 1 1 (x − t)u2 (t)dt. (13.89) e − x− = 4 2 4 0 We ﬁrst set v(x) = u2 (x), u(x) = ± v(x), (13.90) to carry out (13.89) into x 1 1 2x 1 e − x− = (x − t)v(t)dt. 4 2 4 0 Taking the Laplace transform of both sides of (13.91) yields 1 1 1 1 − 2− = 2 V (s), 4(s − 2) 2s 4s s or equivalently 1 , V (s) = s−2 where

(13.91)

(13.92)

(13.93)

406

13 Nonlinear Volterra Integral Equations

V (s) = L{v(x)}.

(13.94)

Taking the inverse Laplace transform of both sides of (13.93) gives v(x) = e2x .

(13.95)

The exact solutions are therefore given by u(x) = ±ex .

(13.96)

It is worth noting that two solutions were obtained because Eq. (13.89) is a nonlinear equation, and the solution may not be unique. Example 13.14 Solve the nonlinear Volterra integral equation of the ﬁrst kind by using the Laplace transform method x 1 1 sin x − x cos x = sin(x − t) sin u(t)dt. (13.97) 2 2 0 We ﬁrst set (13.98) v(x) = sin u(x), u(x) = sin−1 v(x), to carry out (13.97) into x 1 1 sin(x − t)v(t)dt. (13.99) sin x − x cos x = 2 2 0 Taking the Laplace transform of both sides of (13.99) yields 1 1 s2 − 1 = 2 − V (s), (13.100) 2 2 2 2(s + 1) 2(s + 1) s +1 or equivalently 1 . (13.101) V (s) = 2 s +1 Taking the inverse Laplace transform of both sides of the last equation gives v(x) = sin x. (13.102) The exact solution is therefore given by u(x) = x.

(13.103)

Example 13.15 Solve the nonlinear Volterra integral equation of the ﬁrst kind by using the Laplace transform method x 2 1 1 sinh(x − t)u2 (t)dt. (13.104) + cosh 2x − cosh x = 2 6 3 0 We ﬁrst set v(x) = u2 (x), u(x) = ± v(x), (13.105) to carry out (13.104) into x 2 1 1 + cosh 2x − cosh x = sinh(x − t)v(t)dt. 2 6 3 0

(13.106)

13.4 Nonlinear Volterra Integral Equations of the First Kind

Taking the Laplace transform of both sides of (13.106) yields 1 s 2s 1 + − = 2 V (s), 2s 6(s2 − 4) 3(s2 − 1) s −1 or equivalently s 1 . V (s) = − + 2 2s 2(s − 4)

407

(13.107)

(13.108)

Taking the inverse Laplace transform of both sides of the last equation gives v(x) = sinh2 x. The exact solutions are therefore given by

(13.109)

u(x) = ± sinh x.

(13.110)

Example 13.16 Solve the nonlinear Volterra integral equation of the ﬁrst kind by using the Laplace transform method x e2x − ex = ex−tu2 (t)dt. (13.111) 0

We ﬁrst set

v(x) = u2 (x),

to carry out (13.111) into e2x − ex =

u(x) = ±

v(x),

(13.112)

x

ex−tv(t)dt.

(13.113)

0

Taking the Laplace transform of both sides of (13.113) yields 1 . (13.114) V (s) = s−2 Taking the inverse Laplace transform of both sides of the last equation gives v(x) = e2x . The exact solutions are therefore given by

(13.115)

u(x) = ±ex .

(13.116)

Exercises 13.4.1 Use the Laplace transform method to solve the nonlinear Volterra integral equations of the ﬁrst kind: x 1 1 1 4 1. x2 + x3 + (x − t)u2 (t)dt x = 2 3 12 0 x 2 2 2. sin x + cos x − cos(2x) = cos(x − t)u2 (t)dt 3 3 0 x 1 1 ex−t u2 (t)dt 3. e4x − ex = 3 3 0 x 1 2 cos(x − t)u2 (t)dt 4. sin x − sin 2x = 3 3 0

408

13 Nonlinear Volterra Integral Equations

x 1 2 1 1 1 (x − t − 1)u2 (t)dt x − x − sin2 x + sin 2x = 4 2 4 4 0 x 1 1 1 1 5 x = 6. x2 + x3 + x4 + (x − t)u3 (t)dt 2 2 4 20 0 x 1 1 6x 1 e − x− = 7. (x − t)e2u(t) dt 36 6 36 0 x 1 1 ex−t e2u(t) dt 8. e6x − ex = 5 5 0

5.

13.4.2 Conversion to a Volterra Equation of the Second Kind Consider the nonlinear Volterra integral equation of the ﬁrst kind x K(x, t)F (u(t))dt, (13.117) f (x) = 0

where the kernel K(x, t) and the function f (x) are given real-valued functions, and u(x) is the function to be determined. In a manner parallel to our discussion before, we convert (13.117) to a linear Volterra integral equation of the ﬁrst kind of the form x

K(x, t)v(t)dt,

f (x) =

(13.118)

0

by using the transformation This in turn means that

v(x) = F (u(x)).

(13.119)

u(x) = F −1 (v(x)).

(13.120)

We next convert the linear Volterra integral equations of the ﬁrst kind (13.118) to a Volterra integral equations of the second kind. The conversion technique works eﬀectively only if K(x, x) = 0. Diﬀerentiating both sides of (13.118) with respect to x, and using Leibnitz rule, we ﬁnd x Kx (x, t)v(t)dt. (13.121) f (x) = K(x, x)v(x) + 0

Solving for v(x), provided that K(x, x) = 0, we obtain the Volterra integral equation of the second kind given by x f (x) 1 − Kx (x, t)v(t)dt. v(x) = (13.122) K(x, x) 0 K(x, x) Having converted the Volterra integral equation of the ﬁrst kind to the Volterra integral equation of the second kind, we then can use any method that was presented before. Because we solved the Volterra integral equations of the second kind by many methods, we will select distinct methods for solv-

13.4 Nonlinear Volterra Integral Equations of the First Kind

409

ing the nonlinear Volterra integral equation of the ﬁrst kind after converting it to a Volterra integral equation of the second kind. Example 13.17 Convert the nonlinear Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x 1 x −2x )= ex−t u2 (t)dt. (13.123) (e − e 3 0 We ﬁrst set v(x) = u2 (x), u(x) = ± v(x), (13.124) to carry out (13.123) into x 1 x (e − e−2x ) = ex−tv(t)dt. (13.125) 3 0 Diﬀerentiating both sides of (13.125) with respect to x by using Leibnitz rule we ﬁnd the Volterra integral equation of the second kind x 1 ex−t v(t)dt. (13.126) v(x) = (ex + 2e−2x ) − 3 0 This can be solved by many methods as presented in Chapter 3. We select the Laplace transform method to solve this equation. Taking Laplace transform of both sides of (13.126) yields 1 V (s) = . (13.127) s+2 Taking the inverse Laplace transform of both sides gives (13.128) v(x) = e−2x . The exact solutions are therefore given by u(x) = ±e−x .

(13.129)

Example 13.18 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x 1 4 1 3 x + x = (x − t + 1)u2 (t)dt. (13.130) 3 12 0 We ﬁrst set (13.131) v(x) = u2 (x), u(x) = ± v(x), to carry out (13.130) into x 1 1 3 (x − t + 1)v(t)dt. (13.132) x + x4 = 3 12 0 Diﬀerentiating both sides of (13.132) with respect to x by using Leibnitz rule we ﬁnd the Volterra integral equation of the second kind x 1 v(x) = x2 + x3 − v(t)dt. (13.133) 3 0

410

13 Nonlinear Volterra Integral Equations

We will select the modiﬁed decomposition method to solve this equation. Using the recursive relation x 1 3 2 v0 (x) = x , v1 (x) = x − v0 (t)dt = 0. (13.134) 3 0 The exact solutions are therefore given by u(x) = ±x.

(13.135)

Example 13.19 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x 3 3 2x 1 e − x− = (x − t + 1)u2 (t)dt. (13.136) 4 2 4 0 We ﬁrst set (13.137) v(x) = u2 (x), u(x) = ± v(x), to carry out (13.136) into x 3 2x 1 3 (x − t + 1)v(t)dt. e − x− = 4 2 4 0 Diﬀerentiating both sides of (13.138) and proceeding as before we x 3 1 v(x) = e2x − − v(t)dt. 2 2 0 We will select the modiﬁed decomposition method to solve this Using the recursive relation x 1 1 v0 (x) = e2x , v1 (x) = e2x − − v0 (t)dt = 0. 2 2 0 The exact solutions are therefore given by u(x) = ±ex .

(13.138) ﬁnd (13.139) equation. (13.140)

(13.141)

Example 13.20 Convert the Volterra integral equation of the ﬁrst kind to the second kind and solve the resulting equation x 1 1 sinh x + sinh 2x = cosh(x − t)u2 (t)dt. (13.142) 3 3 0 This equation is equivalent to x 1 1 cosh(x − t)v(t)dt, (13.143) sinh x + sinh 2x = 3 3 0 upon setting u2 (x) = v(x). Diﬀerentiating both sides of (13.143) and using Leibnitz rule we obtain x 2 1 sinh(x − t)v(t)dt. (13.144) v(x) = cosh x + cosh(2x) − 3 3 0 For this problem, we select the successive approximations method. We select v0 (x) = 1. Consequently, we obtain

13.5 Systems of Nonlinear Volterra Integral Equations

411

v0 (x) = 1, 2 2 v1 (x) = − cosh x + cosh 2x + 1, 3 3 4 1 4 v2 (x) = − cosh x + cosh 2x + x sinh x, (13.145) 9 9 3 14 11 1 2 14 cosh 2x + x sinh x − x cosh x + 1 v3 (x) = − cosh x + 27 27 36 12 1 4 2 6 2 = 1 + x + x + x + ··· . 3 45 The last approximation is a Taylor series for cosh2 x. This in turn gives (13.146) v(x) = cosh2 x. Consequently, the exact solution of the integral equation is given by u(x) = ± cosh x.

(13.147)

Exercises 13.4.2 Convert the nonlinear Volterra integral equations of the ﬁrst kind to the second kind linear Volterra integral equation, and solve the resulting equation: x 1 1 1 (x − t)u2 (t)dt 1. − x + x2 + sin 2x = 2 2 4 0 x 2 2 2. sin x + cos x − cos 2x = cos(x − t)u2 (t)dt 3 3 0 x 2 1 2 (x − t)u3 (t)dt 3. x − sin x − sin3 x = 3 3 9 0 x 2 1 4. sin x − sin 2x = cos(x − t)u2 (t)dt 3 3 0 x 1 1 2 5 5. x3 − x4 + (x − t)2 u2 (t)dt x = 3 3 15 0 x 1 1 1 6. − − x2 + cosh2 x = (x − t)u2 (t)dt 4 4 4 0 x 1 1 6x 1 e − x− = 7. (x − t)e2u(t) dt 36 6 36 0 x 1 1 ex−t e2u(t) dt 8. e6x − ex = 5 5 0

13.5 Systems of Nonlinear Volterra Integral Equations In this section, systems of nonlinear Volterra integral equations of the ﬁrst kind and the second kind will be studied. Many numerical and analytical methods are usually are used to handle the two kinds of systems. However, in this section we will concern ourselves with the Adomian decomposition

412

13 Nonlinear Volterra Integral Equations

method, and the successive approximations method to handle the nonlinear Volterra integral equations of the ﬁrst kind and the second kind.

13.5.1 Systems of Nonlinear Volterra Integral Equations of the Second Kind Systems of nonlinear Volterra integral equations of the second kind are given by x % 1 (x, t)F%1 (v(t)) dt, u(x) = f1 (x) + K1 (x, t)F1 (u(t)) + K 0 (13.148) x % 2 (x, t)F%2 (v(t)) dt. K2 (x, t)F2 (u(t)) + K v(x) = f2 (x) + 0

The unknown functions u(x) and v(x), that will be determined, occur inside % i (x, t), and the funcand outside the integral sign. The kernels Ki (x, t) and K tion fi (x) are given real-valued functions, for i = 1, 2. The functions Fi and F%i , for i = 1, 2 are nonlinear functions of u(x) and v(x). The Adomian decomposition method, as presented before, decomposes each solution as an inﬁnite sum of components, where these components are determined recurrently. This method can be used in its standard form, or combined with the noise terms phenomenon. Moreover, the modiﬁed decomposition method will be used wherever it is appropriate. It is interesting to point out that the nonlinear functions Fi and F%i , for i = 1, 2, should be replaced by the Adomian polynomials An deﬁned by A0 = F (u0 ), A1 = u1 F (u0 ), 1 A2 = u2 F (u0 ) + u21 F (u0 ), 2! 1 (13.149) A3 = u3 F (u0 ) + u1 u2 F (u0 ) + u31 F (u0 ), 3! 1 2 1 u2 + u1 u3 F (u0 ) + u21 u2 F (u0 ) A4 = u4 F (u0 ) + 2! 2! 1 + u41 F (iv) (u0 ). 4! As stated before, the successive substitutions method, that was used in this chapter and in other chapters as well, will be also used to study the nonlinear systems of the two kinds. Example 13.21 Use the Adomian decomposition method to solve the following system of nonlinear Volterra integral equations

13.5 Systems of Nonlinear Volterra Integral Equations

2 u(x) = x − x3 + 3

x

413

u2 (t) + v(t) dt,

0

(13.150) x 1 v(x) = x2 − x4 + u(t)v(t)dt. 4 0 The Adomian decomposition method suggests that the linear terms u(x) and v(x) be expressed by an inﬁnite series of components ∞ ∞ un (x), v(x) = vn (x), u(x) = (13.151) n=0

n=0

where un (x) and vn (x), n 0 are the components of u(x) and v(x) that will be elegantly determined in a recursive manner. The nonlinear terms u2 (t) and u(t)v(t) are given by the Adomian polynomials An and Bn : A0 = u20 , A1 = 2u0 u1 , A2 = 2u0 u2 + u21 (13.152) and B0 = u0 v0 (t), B1 = v0 (t)u1 (t) + u0 (t)v1 (t), B2 = v0 (t)u2 (t) + u0 (t)v2 (t) + u1 (t)v1 (t),

(13.153)

respectively. Substituting the previous assumptions for linear and nonlinear terms into (13.150), and following Adomian analysis, the system (13.150) is transformed into a set of recursive relations given by x 2 3 (Ak (t) + vk (t)) dt, k 0, (13.154) u0 (x) = x − x , uk+1 (x) = 3 0 and x 1 v0 (x) = x2 − x4 , vk+1 (x) = Bk (t)dt, k 0. (13.155) 4 0 This in turn gives 2 2 19 4 (13.156) u0 (x) = x − x3 , u1 (x) = x3 − x5 + x7 , 3 3 60 63 and 1 1 11 1 (13.157) v0 (x) = x2 − x4 , v1 (x) = x4 − x6 + x8 . 4 4 72 48 It is obvious that the noise terms ∓ 23 x3 appear between u0 (x) and u1 (x). Moreover, the noise terms ∓ 14 x4 appear between v0 (x) and v1 (x). By canceling these noise terms from u0 (x) and v0 (x), the non-canceled terms of u0 (x) and v0 (x) give the exact solutions (u(x), v(x)) = (x, x2 ), that satisfy the given system (13.150).

(13.158)

Example 13.22 Use the Adomian decomposition method to solve the following system of nonlinear Volterra integral equations

414

13 Nonlinear Volterra Integral Equations

1 u(x) = cos x − x2 + 2

x 0

(x − t)(u2 (t) + v 2 (t))dt,

(13.159) x 1 sin2 x + (x − t)(u2 (t) − v 2 (t))dt. 2 0 For this example, we will use the modiﬁed decomposition method, therefore we set the recursive relation u0 (x) = cos x, x (13.160) 1 (x − t)(Ak (t) + Ck (t))dt, k 0, uk+1 (x) = − x2 + 2 0 and v0 (x) = sin x, x (13.161) 1 2 (x − t)(Ak (t) − Ck (t))dt, k 0, vk+1 (x) = − sin x + 2 0 where An (t) and Cn (t) are the Adomian polynomials for u2 (t) and v 2 (t) respectively. This in turn gives u0 (x) = cos x, u1 (x) = 0, uk+1 (x) = 0, k 1, (13.162) v(x) = sin x −

and v0 (x) = sin x,

v1 (x) = 0,

k 1.

vk+1 (x) = 0,

(13.163)

This gives the exact solutions (u(x), v(x)) = (cos x, sin x).

(13.164)

Example 13.23 Use the successive approximations method to solve the following system of nonlinear Volterra integral equations x 2

xu (t) − v 2 (t) dt, u(x) = cos x + sin x + (1 + x) cos2 x − (1 + x2 ) + v(x) = cos x − sin x + (1 − x) cos2 x − (1 + x2 ) +

0 x

u2 (t) + xv 2 (t) dt.

0

(13.165) To use the successive approximations method, we ﬁrst select the zeroth approximations u0 (x) and v0 (x) by (13.166) u0 (x) = v0 (x) = 1. The method of successive approximations admits the use of the iteration formulas un+1 (x) = cos x + sin x + (1 + x) cos2 x − (1 + x2 ) x 2

+ xun (t) − vn2 (t) dt, 0 (13.167) vn+1 (x) = cos x − sin x + (1 − x) cos2 x − (1 + x2 ) x

2 un (t) + xvn2 (t) dt, n 0. + 0

13.5 Systems of Nonlinear Volterra Integral Equations

415

Substituting (13.166) into (13.167) we obtain the series u0 (x) = 1, v0 (x) = 1, 3 7 3 41 5 x + ··· , u1 (x) = 1 + x − x2 − x3 + x4 + 2 6 8 120 3 7 3 41 5 v1 (x) = 1 − x − x2 + x3 + x4 − x + ··· , 2 6 8 120 1 1 13 9 u2 (x) = 1 + x − x2 − x3 − x4 − x5 + · · · , 2 2 8 8 1 1 13 9 v2 (x) = 1 − x − x2 − x3 − x4 + x5 + · · · , 2 2 8 8 1 2 1 3 3 4 89 5 (13.168) x + ··· , u3 (x) = 1 + x − x − x + x + 2 6 8 120 1 1 3 89 5 x + ··· , v3 (x) = 1 − x − x2 + x3 + x4 − 2 6 8 120 1 1 1 1 u4 (x) = 1 + x − x2 − x3 + x4 − x5 + · · · , 2 6 24 8 1 2 1 3 1 4 1 5 v4 (x) = 1 − x − x + x + x + x + · · · , 2 6 24 8 1 2 1 3 1 4 1 u5 (x) = 1 + x − x − x + x + x5 + · · · , 2! 3! 4! 5! 1 1 1 1 v5 (x) = 1 − x − x2 + x3 + x4 − x5 + · · · . 2! 3! 4! 5! This means that the series solutions are given by 1 1 1 1 u(x) = 1 − x2 + x4 + · · · + x − x3 + x5 + · · · , 2! 4! 3! 5! (13.169) 1 4 1 3 1 5 1 2 v(x) = 1 − x + x + · · · − x − x + x + · · · . 2! 4! 3! 5! Consequently, the closed form solutions u(x) and v(x) are given by (u(x), v(x)) = (cos x + sin x, cos x − sin x), (13.170) obtained upon using the Taylor series for cos x and sin x. Notice that this system can be solved easily by using the modiﬁed decomposition method, where we can select u0 (x) = cos x + sin x and v0 (x) = cos x − sin x. Example 13.24 Use the successive approximations method to solve the following system of nonlinear Volterra integral equations x 1 (x − t)(u2 (t) − v 2 (t))dt, u(x) = ex + x − sinh 2x + 2 0 (13.171) x 2 −x x v(x) = e + x − xe + xu (t)v(t)dt. 0

416

13 Nonlinear Volterra Integral Equations

To use the successive approximations method, we ﬁrst select the zeroth approximations u0 (x) and v0 (x) by u0 (x) = v0 (x) = 1. (13.172) The method of successive approximations admits the use of the iteration formulas x 1 x (x − t)(u2n (t) − vn2 (t))dt, un+1 (x) = e + x − sinh(2x) + 2 0 (13.173) x 2 −x x vn+1 (x) = e + x − xe + xun (t)vn (t)dt, n 0. 0

Substituting (13.172) into (13.173) we obtain the series u0 (x) = 1, v0 (x) = 1, 1 1 1 u1 (x) = 1 + x + x2 − x3 + x4 + · · · , 2 2 24 1 2 1 (13.174) v1 (x) = 1 − x + x2 − x3 − x4 + · · · , 2 3 8 1 1 1 u2 (x) = 1 + x + x2 + x3 + x4 + · · · , 2! 3! 4! 1 2 1 3 1 v2 (x) = 1 − x + x − x + x4 − · · · , 2! 3! 4! and so on. Consequently, the solutions u(x) and v(x) are given by (u(x), v(x)) = (ex , e−x ).

(13.175)

Exercises 13.5.1 In Exercises 1–4, use the Adomian decomposition method to solve the systems of nonlinear Volterra integral equations ⎧ x

1 6 1 5 ⎪ ⎪ (x − t)2 u2 (t) − (x − t)v 2 (t) dt ⎪ ⎨ u(x) = x − 30 x + 30 x + 0 1. x ⎪

1 6 1 7 ⎪ ⎪ x − x + (x − t)3 u2 (t) + (x − t)2 v 2 (t) dt ⎩ v(x) = x2 − 60 105 0 ⎧ x

1 ⎪ 2 3 7 ⎪ xtu2 (t) + xtv 2 (t) dt ⎪ ⎨ u(x) = 1 + x − x − 3 x + 0 2. x ⎪

1 ⎪ ⎪ (x − t)u2 (t) − (x − t)v 2 (t) dt ⎩ v(x) = 1 − x2 − x4 + 3 0 ⎧ x

⎪ x x x ⎪ u(x) = 1 + e − 4x(1 − e + xe ) + xtu2 (t) − xtv 2 (t) dt ⎪ ⎨ 0 3. x ⎪

x−t 2 ⎪ ⎪ u (t) − ex−t v 2 (t) dt e ⎩ v(x) = 1 − ex − 4xex + 0

13.5 Systems of Nonlinear Volterra Integral Equations ⎧ x

2 ⎪ x ⎪ − sinh(2x) + u (t) + v 2 (t) dt u(x) = e ⎪ ⎨ 0 4. x ⎪ 2

⎪ ⎪ u (t) − v 2 (t) dt ⎩ v(x) = e−x + 1 − cosh(2x) +

417

0

In Exercises 5–8, use the successive approximations method to solve the systems of nonlinear Volterra integral equations ⎧ x ⎪ ⎪ u(x) = cos x − sin x + cos(x − t)(u2 (t) + v 2 (t))dt ⎪ ⎨ 5.

0

⎪ ⎪ ⎪ ⎩ v(x) = sin x + cos x − x +

x

0

sin(x − t)(u2 (t) + v 2 (t))dt

⎧ x 1 ⎪ ⎪ (u2 (t) + v 2 (t))dt sin(2x) + u(x) = 1 + sin x − 3x + ⎪ ⎨ 2 0 6. x ⎪ 3 2 1 ⎪ 2x+ ⎪ + (x − t)(u2 (t) + v 2 (t))dt v(x) = 1 − sin x − x sin ⎩ 2 2 0 ⎧ x ⎪ ⎪ u(x) = cosh x − x + (u2 (t) − v 2 (t))dt ⎪ ⎨ 0

7.

x ⎪ 1 ⎪ ⎪ (x − t)(u2 (t) + v 2 (t))dt ⎩ v(x) = sinh x − sinh2 x + 2 0 ⎧ x ⎪ ⎪ u(x) = 1 + cosh x − 4 sinh x + (u2 (t) − v 2 (t))dt ⎪ ⎨

8.

⎪ 1 ⎪ ⎪ ⎩ v(x) = 1 − cosh x − sinh(2x) − 3x + 2

0

x

0

(u2 (t) + v 2 (t))dt

13.5.2 Systems of Nonlinear Volterra Integral Equations of the First Kind In this section, we will study a speciﬁc case of the systems of nonlinear Volterra integral equations of the ﬁrst kind given by x % 1 (x, t)F%1 (v(t)) dt, f1 (x) = K1 (x, t)u(t) + K 0 (13.176) x % K2 (x, t)F2 (u(t)) + K2 (x, t)v(t) dt, f2 (x) = 0

% i (x, t), and the functions fi (x) are given where the kernels Ki (x, t) and K real-valued functions, and u(x), and v(x) are the unknown functions that will be determined. Recall that the unknown functions u(x) and v(x) appear inside the integral sign for the Volterra integral equations of the ﬁrst kind. We ﬁrst need to convert this system to a system of nonlinear Volterra integral equation of the second kind. This can be achieved by diﬀerentiating both sides of each part of the system. The conversion technique works eﬀectively by using Leibnitz rule that was presented in section 1.3. Diﬀerentiating both sides of each equation in (13.176), and using Leibnitz rule, we obtain

418

13 Nonlinear Volterra Integral Equations

% 1 (x, x)F%1 (v(x)) f1 (x) = K1 (x, x)u(x) + K x % 1x (x, t)F%1 (v(t)) dt, + K1x (x, t)u(t) + K 0

f2 (x)

% 2 (x, x)v(x) = K2 (x, x)F2 (u(x)) + K x % 2x (x, t)v(t) dt, + K2x (x, t)F2 (u(t)) + K

(13.177)

0

that can be rewritten as % 1 (x, x)F%1 (v(x)) f (x) − K u(x) = 1 K1 (x, x) x 1 % 1x (x, t)F%1 (v(t)) dt, − K1x (x, t)u(t) + K K1 (x, x) 0

(13.178) f2 (x) − K2 (x, x)F2 (u(x)) v(x) = % 2 (x, x) K x 1 % 2x (x, t)v(t) dt. − K2x (x, t)F2 (u(t)) + K % 2 (x, x) 0 K It is obvious that the last system is a system of nonlinear Volterra integral equations of the second kind. This system can be handled by many distinct methods. However, in this section we will use the Adomian decomposition method and the successive approximations method for handling the resulting system of nonlinear Volterra integral equations of the second kind. Notice that other methods can be used as well. The Adomian decomposition method and the successive approximations method were introduced before, hence we skip details. It is important to present the following two remarks:

% 2 (x, x) = 0 for the system to be 1. It is necessary that K1 (x, x) = 0 and K reduced to a system of Volterra integral equations of the second kind. % 2 (x, x) = 0, then we diﬀerentiate again. 2. If K1 (x, x) = 0 and K In the following, we will examine four examples, where Adomian method will be used in the ﬁrst two examples, and the successive approximations method will be used for the other two examples. Example 13.25 Solve the system of nonlinear Volterra integral equations of the ﬁrst kind by using the Adomian method x

1 1 1 3 (x − t + 1)u(t) + (x − t)v 2 (t) dt, x + x4 + x8 = 3 12 56 0 (13.179) x

1 1 1 4 x + x5 + x6 = (x − t)u2 (t) + (x − t + 1)v(t) dt. 4 20 30 0 Diﬀerentiating both sides of each equation, and using Leibnitz rule, we ﬁnd

13.5 Systems of Nonlinear Volterra Integral Equations

1 1 x2 + x3 + x7 = u(x) + 3 7

x

419

(u(t) + v2 (t))dt,

0 x

(13.180) 2

1 1 x3 + x4 + x5 = v(x) + u (t) + v(t) dt. 4 5 0 This system can be rewritten as x

1 1 u(t) + v 2 (t) dt, u(x) = x2 + x3 + x7 − 3 7 0 (13.181) x 2

1 1 u (t) + v(t) dt. v(x) = x3 + x4 + x5 − 4 5 0 Using the standard Adomian decomposition method, we set the recurrence relations 1 1 u0 (x) = x2 + x3 + x7 , 3 7 1 1 v0 (x) = x3 + x4 + x5 , 4 5 x (13.182)

1 1 u0 (t) + v02 (t) dt = − x3 − x7 + · · · , u1 (x) = − 3 7 0 x 2

1 1 v1 (x) = − u0 (t) + v0 (t) dt = − x4 − x5 + · · · . 4 5 0 It is obvious that two noise terms appear between u0 (x) and u1 (x), and two other noise terms between v0 (x) and v1 (x). Canceling the noise terms in u0 (x) and v0 (x) gives the exact solutions by (u(x), v(x)) = (x2 , x3 ). (13.183) Example 13.26 Solve the system of nonlinear Volterra integral equations of the ﬁrst kind by using the modiﬁed Adomian method 1 1 1 1 1 sin x + sin(2x) − sin2 x + x2 − x − 4 4 4 2 4 x

= u(t) + (x − t − 1)v 2 (t) dt, 0 (13.184) 1 2 5 1 2 − cos x − sin x − cos x + x + x + 4 4 4 x

2 = (x − t)u (t) + (x − t + 1)v(t) dt. 0

Diﬀerentiating both sides of each equation, and using Leibnitz rule, we ﬁnd x 1 1 1 1 v2 (t)dt, u(x) = cos x + cos(2x)− sin(2x) + x − − v 2 (x) − 2 4 2 2 0 (13.185) x

2 1 1 v(x) = sin x − cos x + sin(2x) + x + 1 − u (t) + v(t) dt. 4 2 0

420

13 Nonlinear Volterra Integral Equations

To handle this example we select the modiﬁed decomposition method, hence we set the recurrence relations u0 (x) = cos x, v0 (x) = sin x, x 1 1 1 1 v02 (t)dt = 0, u1 (x) = cos(2x) − sin(2x)+ x − −v02 (x) − 2 4 2 2 (13.186) 0 x

1 1 v1 (x) = − cos x + sin(2x) + x + 1 − u20 (t) + v0 (t) dt = 0. 4 2 0 This in turn gives the exact solutions by (u(x), v(x)) = (cos x, sin x).

(13.187)

Example 13.27 Solve the system of nonlinear Volterra integral equations of the ﬁrst kind by using the successive approximations method x

9 1 −2x 1 x (x − t + 1)u(t) + (x − t)v 2 (t) dt, − x− = 2e + e 4 2 4 0 (13.188) x

1 2x 1 1 2 (x − t)u (t) + (x − t + 1)v(t) dt. e + x− = 4 2 4 0 Diﬀerentiating both sides of each equation, and using Leibnitz rule, we ﬁnd x

1 −2x 1 x u(t) + v 2 (t) dt, u(x) = 2e − e − − 2 2 0 (13.189) x

2 1 2x 1 u (t) + v(t) dt. v(x) = e + − 2 2 0 To use the successive approximations method, we ﬁrst select the zeroth approximations u0 (x) and v0 (x) by (13.190) u0 (x) = v0 (x) = 1. The method of successive approximations admits the use of the iteration formulas x

1 1 un+1 (x) = 2ex − e−2x − − un (t) + vn2 (t) dt, 2 2 0 (13.191) x

1 1 vn+1 (x) = e2x + − u2n (t) + vn (t) dt, n 0. 2 2 0 Substituting (13.190) into (13.191) we obtain the series u0 (x) = 1, v0 (x) = 1, 1 u1 (x) = 1 + x + x3 − x4 + · · · , 4 2 1 2 v1 (x) = 1 − x + x + x3 + x4 + · · · , 3 3 1 2 1 4 u2 (x) = 1 + x + x − x + · · · , 2 3

13.5 Systems of Nonlinear Volterra Integral Equations

1 1 v2 (x) = 1 − x + x2 − x4 + · · · , 2 3

421

(13.192)

1 1 u3 (x) = 1 + x + x2 + x3 + · · · , 2 6 1 1 1 v3 (x) = 1 − x + x2 − x3 + x4 + · · · , 2 6 12 u4 (x) = 1 + x +

1 2 1 1 x + x3 + x4 + · · · , 2! 3! 4!

1 2 1 1 x − x3 + x4 − · · · . 2! 3! 4! Consequently, the solutions u(x) and v(x) are given by v4 (x) = 1 − x +

(u(x), v(x)) = (ex , e−x ).

(13.193)

Example 13.28 Solve the system of nonlinear Volterra integral equations of the ﬁrst kind by using the successive approximations method 1 1 1 −2 cos x + sin(2x) + x2 + x + 2 4 2 2 x

(x − t + 1)u(t) + (x − t)v 2 (t) dt, = 0 (13.194) 1 1 1 2 sin x − sin(2x) + x2 − x 4 2 2 x

= (x − t)u2 (t) + (x − t + 1)v(t) dt. 0

Diﬀerentiating both sides of each equation, and using Leibnitz rule, we ﬁnd x

1 1 u(x) = 2 sin x + cos(2x) + x + − u(t) + v 2 (t) dt, 2 2 0 (13.195) x

2 1 1 v(x) = 2 cos x − cos(2x) + x − − u (t) + v(t) dt. 2 2 0 We select the zeroth approximations u0 (x) and v0 (x) by u0 (x) = v0 (x) = 1. We next use the iteration formulas x

1 1 un (t) + vn2 (t) dt, un+1 (x) = 2 sin x + cos(2x) + x + − 2 2 0 x

2 1 1 un(t) + vn (t) dt, n 0. vn+1 (x) = 2 cos x − cos(2x) + x − − 2 2 0 (13.196) Substituting the zeroth selections into (13.196) we obtain the series

422

13 Nonlinear Volterra Integral Equations

u0 (x) = 1,

v0 (x) = 1, 1 1 1 u1 (x) = 1 + x − x2 − x3 + x4 + x5 + · · · , 3 3 60 1 v1 (x) = 1 − x − x4 + · · · , 4 1 1 5 1 u2 (x) = 1 + x − x2 − x3 + x4 + x5 + · · · , 2 3 12 20 1 2 1 3 5 4 3 v2 (x) = 1 − x − x + x + x − x5 + · · · , 2 3 12 20 1 1 3 u3 (x) = 1 + x − x2 − x3 − x5 + · · · , (13.197) 2 6 20 1 2 1 3 1 4 1 5 v3 (x) = 1 − x − x + x + x − x + · · · , 2 6 12 6 1 2 1 3 1 4 u4 (x) = 1 + x − x − x + x + · · · , 2 6 24 1 2 1 3 1 v4 (x) = 1 − x − x + x + x4 + · · · , 2 6 24 1 1 1 1 u5 (x) = 1 + x − x2 − x3 + x4 + x5 + · · · , 2! 3! 4! 5! 1 2 1 3 1 4 1 v5 (x) = 1 − x − x + x + x − x5 + · · · . 2! 3! 4! 5! This means that the series solutions are given by 1 4 1 3 1 5 1 2 u(x) = 1 − x + x + · · · + x − x + x + · · · , 2! 4! 3! 5! (13.198) 1 2 1 4 1 3 1 5 v(x) = 1 − x + x + · · · − x − x + x + · · · . 2! 4! 3! 5! Consequently, the closed form solutions u(x) and v(x) are given by (u(x), v(x)) = (cos x + sin x, cos x − sin x). (13.199)

Exercises 13.5.2 In Exercises 1–4, use the Adomian decomposition method to solve the following systems of Volterra integral equations of the ﬁrst kind ⎧ x

1 2 1 5 1 6 ⎪ ⎪ x x x u(t) + (x − t + 1)v 2 (t) dt + + = ⎪ ⎨2 5 30 0 1. x ⎪

1 1 ⎪ ⎪ (x − t)u2 (t) + (x − t + 1)v(t) dt ⎩ x3 + x4 = 3 6 0 ⎧ x

⎪ 1 x4 + 1 x5 + 1 x12 = ⎪ (x − t + 1)u(t) + (x − t)v 2 (t) dt ⎪ ⎨4 20 132 0 2. x ⎪

1 1 1 ⎪ 6 7 8 ⎪ x + x = (x − t)u2 (t) + (x − t + 1)v(t) dt ⎩ x + 6 42 56 0

References 423 ⎧ x

1 2 3 1 ⎪ 2 ⎪ u(t) − (x − t)v 2 (t) dt ⎪ ⎨ − cos x + 4 cos x − 4 x + 4 = 0 3. x ⎪

1 1 3 ⎪ ⎪ (x − t)u2 (t) + (x − t + 1)v(t) dt ⎩ sin x − cos x + cos2 x + x2 + = 4 4 4 0 ⎧ x

1 2 1 1 ⎪ 2 ⎪ u(t) − (x − t)v 2 (t) dt ⎪ ⎨ sinh x − 4 cosh x + 4 x + 4 = 0 4. x ⎪

1 5 1 ⎪ ⎪ (x − t)u2 (t)v(t) dt ⎩ cosh x + cosh2 x + x2 − = 4 4 4 0 In Exercises 5–8, use the successive approximations method to solve the systems of Volterra integral equations of the ﬁrst kind ⎧ x

⎪ 2 ⎪ x + 2x = 1 − 2 cos x + cos (x − t + 1)u(t) + v 2 (t) dt ⎪ ⎨ 0 5. x ⎪ 2

⎪ ⎪ u (t) + (x − t + 1)v(t) dt ⎩ 1 − 2 sin x − cos2 x + 2x = 0

⎧ x

1 3 2x 1 −4x 1 ⎪ ⎪ − − e e x − = (x − t + 1)u(t) + v 2 (t) dt ⎪ ⎨4 4 2 2 0 6. x ⎪

1 1 1 ⎪ ⎪ u2 (t) + (x − t + 1)v(t) dt ⎩ e4x − e−4x + x = 4 4 2 0 ⎧ x

1 2x 1 2 1 ⎪ ⎪ e + x +x− = (x − t + 1)u(t) + v 2 (t) dt ⎪ ⎨2 2 2 0 7. x ⎪ 2

1 1 1 ⎪ ⎪ u (t) + (x − t + 1)v(t) dt ⎩ e2x + x2 + 3x − = 2 2 2 0 ⎧ x

5 2 3 1 ⎪ 2 ⎪ x + + cos x = (x − t + 1)u(t) + (x − t)v 2 (t) dt sin x − cos x + ⎪ ⎨ 4 4 4 0 8. x ⎪

1 5 5 ⎪ ⎪ (x − t)u2 (t) + (x − t + 1)v(t) dt ⎩ − sin x + cos x+ cos2 x+ x2 + 2x − = 4 4 4 0

References 1. H.T. Davis, Introduction to Nonlinear Diﬀerential and Integral Equations, Dover, Publications, New York, (1962). 2. A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 3. P. Linz, A simple approximation method for solving Volterra integro-diﬀerential equations of the ﬁrst kind, J. Inst. Math. Appl., 14 (1974) 211–215. 4. P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, (1985). 5. R.K. Miller, Nonlinear Volterra Integral Equations, W.A. Benjamin, Menlo Park, CA, (1967). 6. W.E. Olmstead and R.A. Handelsman, Asymptotic solution to a class of nonlinear Volterra integral equations, II, SIAM J. Appl. Math., 30 (1976) 180–189. 7. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997). 8. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).

Chapter 14

Nonlinear Volterra Integro-Diﬀerential Equations

14.1 Introduction It is well known that linear and nonlinear Volterra integral equations arise in many scientiﬁc ﬁelds such as the population dynamics, spread of epidemics, and semi-conductor devices. Volterra started working on integral equations in 1884, but his serious study began in 1896. The name integral equation was given by du Bois-Reymond in 1888. The linear Volterra integro-diﬀerential equations were presented in Chapter 5. It is our goal in this chapter to study the nonlinear Volterra integrodiﬀerential equations of the ﬁrst and the second kind. The nonlinear Volterra integro-diﬀerential equations are characterized by at least one variable limit of integration. The nonlinear Volterra integro-diﬀerential equation of the second kind reads x K(x, t)F (u(t)) dt, (14.1) u(n) (x) = f (x) + 0

and the standard form of the nonlinear Volterra integro-diﬀerential equation [1–3] of the ﬁrst kind is given by x x K1 (x, t)F (u(t))dt + K2 (x, t)u(n) (t)dt = f (x), K2 (x, x) = 0, (14.2) 0

0

where u(n) (x) is the n th derivative of u(x). For these equations, the kernels K(x, t), K1 (x, t) and K2 (x, t), and the function f (x) are given realvalued functions. The function F (u(x)) is a nonlinear function of u(x) such as u2 (x), sin(u(x)), and eu(x) .

A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

426

14 Nonlinear Volterra Integro-Diﬀerential Equations

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind The linear Volterra integro-diﬀerential equation [4–6], where both diﬀerential and integral operators appear together in the same equation, has been studied in Chapter 5. In this section, we will extend the work presented in Chapter 5 to nonlinear Volterra integro-diﬀerential equation. The nonlinear Volterra integro-diﬀerential equation of the second kind reads x u(i) (x) = f (x) + K(x, t)F (u(t))dt, (14.3) 0 i

where u(i) (x) = ddxui , and F (u(x)) is a nonlinear function of u(x). Because the equation in (14.3) combines the diﬀerential operator and the integral operator, then it is necessary to deﬁne initial conditions for the determination of the particular solution u(x) of the nonlinear Volterra integro-diﬀerential equation. The nonlinear Volterra integro-diﬀerential equation [7–10] appeared after its establishment by Volterra. It appears in many physical applications such as glass-forming process, heat transfer, diﬀusion process in general, neutron diﬀusion and biological species coexisting together with increasing and decreasing rates of generating. More details about the sources where these equations arise can be found in physics, biology and books of engineering applications. In Chapter 5, we applied many methods to handle the linear Volterra integro-diﬀerential equations of the second kind. In this section we will use only some of these methods. However, the other methods presented in Chapter 5 can be used as well. In what follows we will apply the combined Laplace transform-Adomian decomposition method, the variational iteration method (VIM), and the series solution method to handle nonlinear Volterra integrodiﬀerential equations of the second kind (14.3).

14.2.1 The Combined Laplace Transform-Adomian Decomposition Method In this section we will consider the kernel K(x, t) of (14.3) as a diﬀerence kernel that depends on the diﬀerence x − t, such as ex−t, cosh(x − t), and sin(x − t). The nonlinear Volterra integro-diﬀerential equation (14.3) can thus be expressed as x u(i) (x) = f (x) + K(x − t)F (u(t)) dt. (14.4) 0

To solve the nonlinear Volterra integro-diﬀerential equations by using the Laplace transform method, it is essential to use the Laplace transforms of

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

427

the derivatives of u(x). We can easily show that L{u(i) (x)} = si L{u(x)} − si−1 u(0) − si−2 u (0) − · · · − u(i−1) (0). This simply gives L{u (x)} = sL{u(x)} − u(0) = sU (s) − u(0), L{u (x)} = s2 L{u(x)} − su(0) − u (0) = s2 U (s) − su(0) − u (0), L{u (x)} = s3 L{u(x)} − s2 u(0) − su (0) − u (0) = s3 U (s) − s2 u(0) − su (0) − u (0),

(14.5)

(14.6)

L{u(iv) (x)} = s4 L{u(x)} − s3 u(0) − s2 u (0) − su (0) − u (0) = s4 U (s) − s3 u(0) − s2 u (0) − su (0) − u (0), and so on for derivatives of higher order, where U (s) = L{u(x)}. Applying the Laplace transform to both sides of (14.4) gives si L{u(x)} − si−1 u(0) − si−2 u (0) − · · · − u(i−1) (0) (14.7) = L{f (x)} + L{K(x − t)}L{F (u(t))}, or equivalently 1 1 1 L{u(x)} = u(0) + 2 u (0) + · · · + i u(i−1) (0) s s s (14.8) 1 1 + i L{f (x)} + i L{K(x − t)}L{F (u(t))}. s s To overcome the diﬃculty of the nonlinear term F (u(x)), we apply the Adomian decomposition method for handling (14.8). To achieve this goal, we ﬁrst represent the linear term u(x) at the left side by an inﬁnite series of components given by ∞ u(x) = un (x), (14.9) n=0

where the components un (x), n 0 will be recursively determined. However, the nonlinear term F (u(x)) at the right side of (14.8) will be represented by an inﬁnite series of the Adomian polynomials An in the form n ∞ 1 dn i An (x), An = λ ui , n = 0, 1, 2, · · · , F F (u(x)) = n! dλn n=0 i=0 λ=0 (14.10) where An , n 0 can be obtained for all forms of nonlinearity. Substituting (14.9) and (14.10) into (14.8) leads to + ∞ 1 1 1 un (x) = u(0) + 2 u (0) + · · · + i u(i−1) (0) L s s s n=0 ∞ + (14.11) 1 1 + i L{f (x)} + i L{K(x − t)}L An (x) . s s n=0

428

14 Nonlinear Volterra Integro-Diﬀerential Equations

The Adomian decomposition method admits the use of the following recursive relation 1 1 1 1 L{u0 (x)} = u(0) + 2 u (0) + · · · + i u(i−1) (0) + i L{f (x)}, s s s s (14.12) 1 L{uk+1 (x)} = i L{K(x − t)}L{Ak (x)}, k 0. s The necessary conditions presented in Chapter 1 for Laplace transform method concerning the limit as s → ∞, should be satisﬁed here for a successful use of this method. Applying the inverse Laplace transform to the ﬁrst part of (14.12) gives u0 (x), that will deﬁne A0 . This in turn will lead to the complete determination of the components of uk+1 , k 0 upon using the second part of (14.12). The combined Laplace transform Adomian-decomposition method for solving nonlinear Volterra integro-diﬀerential equations of the second kind will be illustrated by studying the following examples. Example 14.1 Solve the nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method x 1 2 9 5 1 −2x −x u (x) = − x − x − 3e − e + (x − t)u2 (t) dt, u(0) = 2. (14.13) 4 2 2 4 0 Notice that the kernel K(x − t) = (x − t). Taking Laplace transform of both sides of (14.13) gives 1 2 9 5 1 −2x −x − x − x − 3e − e L{u (x)} = L +L{(x−t)∗u2 (x)}, (14.14) 4 2 2 4 so that 5 1 1 9 1 3 − − + L{u2 (x)}, (14.15) sU (s) − u(0) = − 3− 4s 2s2 s s + 1 4(s + 2) s2 or equivalently 9 1 1 5 1 3 2 − + 3 L{u2 (x)}. (14.16) U (s) = + 2 − 3 − 4 − s 4s 2s s s(s + 1) 4s(s + 2) s Substituting the series assumption for U (s) and the Adomian polynomials for u2 (x) as given above in (14.9) and (14.10) respectively, and using the recursive relation (14.12) we obtain 2 5 1 3 9 1 − , U0 (s) = + 2 − 3 − 4 − s 4s 2s s s(s + 1) 4s(s + 2) (14.17) 1 L{uk+1 (x)} = 3 L{Ak (x)}, k 0. s Recall that the Adomian polynomials for F (u(x)) = u2 (x) are given by A0 = u20 , A1 = 2u0 u1 , A2 = 2u0 u2 + u21 , (14.18) A3 = 2u0 u3 + 2u1 u2 .

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

429

Taking the inverse Laplace transform of both sides of the ﬁrst part of (14.17), and using the recursive relation (14.17) gives 1 5 5 7 5 u0 (x) = 2 − x + x2 − x3 + x4 − x + ··· , 2 6 24 120 (14.19) 2 1 1 u1 (x) = x3 − x4 + x5 + · · · . 3 6 20 Using (14.9), the series solution is therefore given by 1 1 1 1 (14.20) u(x) = 2 − x + x2 − x3 + x4 − x5 + · · · , 2! 3! 4! 5! that converges to the exact solution u(x) = 1 + e−x . (14.21) Example 14.2 Solve the nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method x 2 cos(x − t)u2 (t)dt, u(0) = 1. (14.22) u (x) = − (2 sin x + sin(2x)) + 3 0 Notice that the kernel K(x − t) = cos(x − t). Taking Laplace transform of both sides of (14.22) gives 2 L{u (x)} = L − (2 sin x + sin(2x)) + L{cos(x − t) ∗ u2 (x)}, (14.23) 3 so that 4 2 s (14.24) sU (s) − u(0) = − 2 − + 2 L{u2 (x)}, 2 3(s + 1) 3(s + 4) s + 1 or equivalently 4 2 1 1 − + 2 L{u2 (x)}. U (s) = − (14.25) 2 2 s 3s(s + 1) 3s(s + 4) s + 1 Substituting the series assumption for U (s) and the Adomian polynomials for u2 (x) as given above in (14.9) and (14.10) respectively, and using the recursive relation (14.12) we obtain 2 5 1 3 9 1 − , U0 (s) = + 2 − 3 − 4 − s 4s 2s s s(s + 1) 4s(s + 2) (14.26) 1 L{uk+1 (x)} = 3 L{Ak (x)}, k 0. s Taking the inverse Laplace transform of both sides of the ﬁrst part of (14.26), and using the recursive relation (14.26) gives 1 1 u0 (x) = 1 − x2 + x4 − x6 + · · · , 6 60 1 5 37 6 u1 (x) = x2 − x4 + x + ··· , (14.27) 2 24 720 1 4 1 x − x6 + · · · , u2 (x) = 22 20

430

14 Nonlinear Volterra Integro-Diﬀerential Equations

1 6 x + ··· . 72 Using (14.9), the series solution is therefore given by 1 1 1 u(x) = 1 − x2 + x4 − x6 + · · · , 2! 4! 6! that converges to the exact solution u(x) = cos x. u3 (x) =

(14.28) (14.29)

Example 14.3 Solve the nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method x 2 2 x 2x ex−t u2 (t)dt, u(0) = 1, u (0) = 2. u (x) = 2 + 2x + x − x e − e + 0

(14.30) Notice that the kernel K(x − t) = ex−t . Taking Laplace transform of both sides of (14.30) gives (14.31) L{u (x)} = L{2 + 2x + x2 − x2 ex − e2x } + L{ex−t ∗ u2 (x)}, so that s2 U (s) − su(0) − u (0) =

2 1 2 2 2 1 + + L{u2 (x)}, + 3 − − s s2 s (s − 1)3 s − 2 s − 1 (14.32)

or equivalently 2 2 2 2 1 1 2 1 − 2 U (s) = + 2 + 3 + 4 + 5 − 2 + 2 L{u2 (x)}. 3 s s s s s s (s − 1) s (s − 2) s (s − 1) (14.33) Proceeding as before we ﬁnd 2 1 2 2 2 2 1 U0 (s) = + 2 + 3 + 4 + 5 − 2 , − 2 3 s s s s s s (s − 1) s (s − 2) (14.34) 1 L{uk+1 (x)} = 2 L{Ak (x)}, k 0. s (s − 1) Taking the inverse Laplace transform of both sides of the ﬁrst part of (14.34), and using the recursive relation (14.34) gives 1 1 7 7 6 x + ··· , u0 (x) = 1 + 2x + x2 − x4 − x5 − 2 6 60 180 1 5 1 3 (14.35) u1 (x) = x3 + x4 + x5 + x6 + · · · , 6 24 8 80 1 6 u2 (x) = x + ··· . 360 Using (14.9), the series solution is therefore given by 1 2 1 3 1 4 1 5 1 6 u(x) = x + 1 + x + x + x + x + x + x + · · · , (14.36) 2! 3! 4! 5! 6! that converges to the exact solution

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

u(x) = x + ex .

431

(14.37)

Example 14.4 Solve the nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method x 1 sin(x − t)u2 (t)dt, (14.38) u (x) = −1 − (sin x + sin(2x)) + 2 cos x + 3 0 where u(0) = −1, u (0) = 1. Notice that the kernel K(x − t) = sin(x − t). Taking Laplace transform of both sides of (14.38) gives 1 L{u (x)} = L −1 − (sin x + sin(2x)) + 2 cos x + L{sin(x − t) ∗ u2 (x)}, 3 (14.39) so that 1 2 2s 1 1 s2 U (s)−su(0)−u (0) = − − 2 − + + L{u2 (x)}, s 3(s + 1) 3(s2 + 4) s2 + 1 s2 + 1 (14.40) so that 1 2 2 1 1 1 − + U (s) = − + 2 − 3 − 2 2 s s s 3s (s + 1) 3s2 (s2 + 4) s(s2 + 1) 1 + 2 2 (14.41) L{u2 (x)}. s (s + 1) Proceeding as before we ﬁnd 1 1 1 1 1 6 11 7 u0 (x) = −1 + x + x2 − x3 − x4 + x5 + x − x + ··· , 2 6 12 40 360 5040 1 4 1 1 6 1 7 x − x5 − x + x + ··· . (14.42) u1 (x) = 24 60 720 504 Using (14.9), the series solution is therefore given by 1 1 1 1 1 1 u(x) = x − x3 + x5 − x7 + · · · − 1 − x2 + x4 − x6 + · · · , 3! 5! 7! 2! 4! 6! (14.43) that converges to the exact solution u(x) = sin x − cos x. (14.44)

Exercises 14.2.1 Solve the following nonlinear Volterra integro-diﬀerential equations by using the combined Laplace transform-Adomian decomposition method x 17 1 9 1. u (x) = (x − t)u2 (t)dt, u(0) = 3 + x − 2x2 − 3ex − e2x + 4 2 4 0 x 11 1 3 1 4 (x − t)u2 (t)dt, u(0) = 1 − x− x + ex (5 − 2x) − e2x + 2. u (x) = − 4 2 12 4 0

432 3. 4. 5. 6.

14 Nonlinear Volterra Integro-Diﬀerential Equations x 1 1 u (x) = 1 − ex + e−2x + ex−t u2 (t)dt, u(0) = 0 3 3 0 x 5 1 1 u (x) = − + cos x − cos2 x + sin(x − t)u2 (t)dt, u(0) = 0 3 3 3 0 x 1 5 u (x) = − sin x + sin(2x) + cos(x − t)u2 (t)dt, u(0) = 0, u (0) = 1 3 3 0 x 1 1 cosh(x − t)u2 (t)dt, u (x) = cosh x − sinh x − sinh(2x) + 3 3 0 u(0) = 1, u (0) = 0

7. u (x) =

3 x 1 3x e − e + 2 2

8. u (x) = −

x

0

ex−t u3 (t)dt, u(0) = u (0) = u (0) = 1

2 5 4 − cos x + cos2 x + 3 3 3

x

0

cos(x − t)u2 (t)dt,

u(0) = u (0) = 1, u (0) = −1

14.2.2 The Variational Iteration Method The variational iteration method was used before in other chapters. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the Adomian polynomials. The standard i th order nonlinear Volterra integro-diﬀerential equation is of the form x u(i) (x) = f (x) + K(x, t)F (u(t))dt, (14.45) 0 (i)

di u dxi ,

and F (u(x)) is a nonlinear function of u(x). The initial where u (x) = conditions should be prescribed for the complete determination of the exact solution. The correction functional for the nonlinear integro-diﬀerential equation (14.45) is x

un+1 (x) = un (x) +

0

λ(ξ) u(i) n (ξ) − f (ξ) −

ξ

0

K(ξ, r)F (˜ un (r)) dr

dξ.

(14.46) The variational iteration method is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ that can be identiﬁed optimally via integration by parts and by using a restricted variation. The Lagrange multiplier λ may be a constant or a function. Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), n 0 of the solution

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

433

u(x). The zeroth approximation u0 can be any selective function. However, the initial values u(0), u (0), . . . are preferably used for the selective zeroth approximation u0 . In what follows we summarize the Lagrange multipliers as derived in Chapter 3, and the selective zeroth approximations: u + f (u(ξ), u (ξ)) = 0, λ = −1, u0 (x) = u(0), u + f (u(ξ), u (ξ), u (ξ)) = 0, λ = ξ − x, u0 (x) = u(0) + u (0)x 1 (14.47) u + f (u(ξ), u (ξ), u (ξ), u (ξ)) = 0, λ = − (ξ − x)2 , 2! 1 u0 (x) = u(0) + u (0)x + u (0)x2 , 2! and so on. Consequently, the solution is given by u(x) = lim un (x). (14.48) n→∞

The VIM will be illustrated by studying the following examples. Example 14.5 Use the variational iteration method to solve the nonlinear Volterra integrodiﬀerential equation x x x 2x ex−t u2 (t)dt, u(0) = 2. (14.49) u (x) = 1 + e − 2xe − e + 0

The correction functional for this equation is given by x t un (t) − 1 − et + 2tet + e2t − un+1 (x) = un (x) − et−r u2n(r) dr dt, 0

0

(14.50) where we used λ = −1 for ﬁrst-order integro-diﬀerential equation. We can use the initial condition to select u0 (x) = u(0) = 2. Using this selection into the correction functional gives the following successive approximations u0 (x) = 2, 1 1 3 19 5 x + ··· , u1 (x) = 2 + x + x2 − x3 − x4 − 2 2 8 120 (14.51) 1 1 1 1 u2 (x) = 2 + x + x2 + x3 + x4 − x5 + · · · , 2! 3! 4! 8 1 2 1 3 1 4 1 u3 (x) = 2 + x + x + x + x + x5 + · · · , 2! 3! 4! 5! and so on for other approximations. The VIM admits the use of u(x) = lim un (x), n→∞

that gives the exact solution by u(x) = 1 + ex .

(14.52) (14.53)

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14 Nonlinear Volterra Integro-Diﬀerential Equations

Example 14.6 Use the variational iteration method to solve the nonlinear Volterra integrodiﬀerential equation x u (x) = −x + sec x tan x − tan x + (1 + u2 (t)) dt, u(0) = 1. (14.54) 0

The correction functional for this equation is given by un+1 (x) = un (x) x t un (t) + t − sec t tan t + tan t − − (1 + u2n (r))dr dt, 0

0

(14.55) where we used λ = −1 for ﬁrst-order integro-diﬀerential equation. We can use the initial condition to select u0 (x) = u(0) = 1. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1, 1 1 1 u1 (x) = 1 + x2 + x4 + x6 + · · · , 2 8 16 (14.56) 1 2 5 4 19 6 u2 (x) = 1 + x + x + x + ··· , 2 24 240 1 2 5 4 61 6 x + ··· , u3 (x) = 1 + x + x + 2 24 720 and so on. The VIM admits the use of (14.57) u(x) = lim un (x), n→∞

that gives the exact solution by u(x) = sec x.

(14.58)

Example 14.7 Use the variational iteration method to solve the nonlinear Volterra integrodiﬀerential equation x u (x) = x + cos x − tan x + tan2 x + (sin t + u2 (t)) dt, u(0) = 0. (14.59) 0

The correction functional for this equation is given by un+1 (x) = un (x) t x (sin t + u2n (r))dr dt. un(t) − t − cos t + tan t − tan2 t − − 0

0

(14.60) We can use the initial condition to select u0 (x) = u(0) = 0. Using this selection into the correction functional gives the following successive approximations

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

435

u0 (x) = 0, 1 1 2 1 17 7 x + ··· , u1 (x) = x + x3 − x4 + x5 − x6 + 3 12 15 45 315 1 2 17 7 u2 (x) = x + x3 + x5 + x + ··· , 3 15 315 and so on. The VIM admits the use of u(x) = lim un (x), n→∞

that gives the exact solution by u(x) = tan x.

(14.61)

(14.62) (14.63)

Example 14.8 Use the variational iteration method to solve the nonlinear Volterra integrodiﬀerential equation x 1 5 u (x) = − sin x + sin(2x) + cos(x − t)u2 (t)dt, u(0) = 0, u (0) = 1. 3 3 0 (14.64) The correction functional for this equation is given by un+1 (x) = un (x) t x 1 5 2 (t − x) un (t) + sin t − sin(2t) − cos(t − r)un (r)dr dt. + 3 3 0 0 (14.65) We can use the initial condition to select u0 (x) = u(0) + xu (0) = x. Using this selection into the correction functional gives the following successive approximations u0 (x) = x, 1 1 1 7 x + ··· , u1 (x) = x − x3 + x5 + (14.66) 3! 5! 720 1 1 1 u2 (x) = x − x3 + x5 − x7 + · · · , 3! 5! 7! and so on. The VIM gives the exact solution by u(x) = sin x.

(14.67)

Exercises 14.2.2 Solve the following nonlinear Volterra integro-diﬀerential equations by using the variational iteration method x 1 1 1. u (x) = (1 − x2 ) − sin x − cos2 x + (x − t)(1 − u2 (t))dt, u(0) = 0 4 4 0 x 1 1 (x − t)(1 − u2 (t))dt, u(0) = 1 2. u (x) = − (1 − 2x + 2x2 ) − e−x + e−2x + 4 4 0 x 2 2 xte−u (t)dt, u(0) = 0 3. u (x) = 1 + u − xe−x − 2 0

436

14 Nonlinear Volterra Integro-Diﬀerential Equations x 4 1 4. u (x) = − sin x − sin(2x) + cos(x − t)u2 (t)dt, u(0) = 1 3 3 0 x 1 1 5. u (x) = − x − cos x + sin(2x) + (1 − u2 (t))dt, u(0) = 1, u (0) = 0 2 4 0 x 6. u (x) = 1 + ex − e2x + ex−t (1 + u2 (t))dt, u(0) = 1, u (0) = 1 0

x

7. u (x) = e (2 − x) − e

2x

x

+ 0

ex−t (u + u2 (t))dt, u(0) = 1, u (0) = 1

8. u (x) = −x + tan x(6 sec3 x − sec x + 1) +

x

0

(1 − u2 (t))dt,

u(0) = 1, u (0) = 0, u (0) = 1

14.2.3 The Series Solution Method The series solution method [5,7] was eﬀectively used in this text to handle integral and integro-diﬀerential equations. The method stems mainly from the Taylor series for analytic functions. A real function u(x) is called analytic if it has derivatives of all orders such that the Taylor series at any point b in its domain k u(n) (b) (x − b)n , uk (x) = (14.68) n! n=0 converges to u(x) in a neighborhood of b. For simplicity, the generic form of Taylor series at x = 0 can be written as ∞ a n xn . (14.69) u(x) = n=0

The Taylor series method, or simply the series solution method will be used in this section for solving nonlinear Volterra integro-diﬀerential equations. We will assume that the solution u(x) of the nonlinear Volterra integrodiﬀerential equation x u(n) (x) = f (x) + λ K(x, t)F (u(t))dt, u(k) (0) = k!ak , 0 k (n − 1), 0

(14.70) is analytic, and therefore possesses a Taylor series of the form given in (14.69), where the coeﬃcients an will be determined recurrently. The ﬁrst few coeﬃcients ak can be determined by using the initial conditions so that 1 1 (14.71) a0 = u(0), a1 = u (0), a2 = u (0), a3 = u (0), 2! 3! and so on. The remaining coeﬃcients ak of (14.69) will be determined by applying the series solution method to the nonlinear Volterra integro-diﬀerential equation (14.70). Substituting (14.69) into both sides of (14.70) gives

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

∞

(n) ak x

k

x

= T (f (x)) +

K(x, t)F 0

k=0

∞

437

ak t

k

dt,

(14.72)

k=0

where T (f (x)) is the Taylor series for f (x). The integro-diﬀerential equation (14.70) will be converted to a traditional integral in (14.72) where instead of integrating the unknown function F (u(x)), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integral in (14.72), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0, where some of these coeﬃcients will be used from the initial conditions. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (14.69). Example 14.9 Solve the nonlinear Volterra integro-diﬀerential equation by using the series solution method x x 2x u (x) = 1 − e + e + ex−t (1 − u2 (t))dt, u(0) = 1. (14.73) 0

Substituting u(x) by the series u(x) =

∞

an xn ,

(14.74)

n=0

into both sides of equation (14.73) leads to ⎛ ⎛ ∞ ∞ 2 ⎞⎞ x ⎝T2 (ex−t ) ⎝1 − an xn = T1 (1 − ex + e2x ) + an tn ⎠⎠ dt, n=0

0

n=0

(14.75) where T1 and T2 are the Taylor series about x = 0 and about t = 0 respectively. Evaluating the integral at the right side, using a0 = 1, we ﬁnd a1 + 2a2 x + 3a3 x2 + 4a4 x3 + 5a5 x4 + · · · 3 1 7 1 2 2 − a1 x2 + − a1 − a2 − a1 x3 =1+x+ 2 6 3 3 3 1 2 1 5 1 1 1 − a − a2 − a1 − a3 − a1 a2 x4 + 8 12 1 6 12 2 2 1 31 1 1 1 2 2 1 1 − a1 − a21 − a2 − a22 − a4 − a1 a3 − a3 − a1 a2 x5 + 120 60 60 30 5 5 5 10 10 +O(x6 ). (14.76)

438

14 Nonlinear Volterra Integro-Diﬀerential Equations

Equating the coeﬃcients of like powers of x in both sides, and solving the system of equations we obtain 1 an = , n 0. (14.77) n! This gives the exact solution by (14.78) u(x) = ex . Example 14.10 Solve the nonlinear Volterra integro-diﬀerential equation by using the series solution method x 5 1 cos(x − t)(1 + u2 (t))dt, u(0) = 0. u (x) = cos x − sin x + sin(2x) + 3 3 0 (14.79) Substituting u(x) by the series ∞ u(x) = an xn , (14.80) n=0

into both sides of equation (14.79) leads to ∞ 5 1 n an x = T1 cos x − sin x + sin(2x) 3 3 n=0 ⎛ ∞ 2 ⎞ x (T2 (cos(x − t)) ⎝1 + an tn ⎠ dt, + 0

(14.81)

n=0

where T1 and T2 are the Taylor series about x = 0 and about t = 0 respectively. Evaluating the integral at the right side, using a0 = 0, and proceeding as before we ﬁnd (−1)n (14.82) a2n+1 = , a2n = 0, n 0. (2n + 1)! This gives the exact solution by u(x) = sin x.

(14.83)

Example 14.11 Solve the nonlinear Volterra integro-diﬀerential equation by using the series solution method x 1 1 u (x) = x−sin x−cos x− sin(2x)+ (x−t)(1−u2(t))dt, u(0) = u (0) = 1. 2 4 0 (14.84) Substituting u(x) by the series ∞ an xn , (14.85) u(x) = n=0

into both sides of equation (14.84) leads to

14.2 Nonlinear Volterra Integro-Diﬀerential Equations of the Second Kind

∞

n

an x

n=0

439

1 1 x − sin x − cos x − sin(2x) 2 4 ⎛ ⎛ 2 ⎞⎞ x ∞ ⎝(x − t) ⎝1 − an tn ⎠⎠ dt, +

= T1

0

(14.86)

n=0

where T1 is the Taylor series about x = 0. Evaluating the integral at the right side, using a0 = 1, a1 = 1, and proceeding as before we ﬁnd (−1)n (−1)n , a2n+1 = , n 0. (14.87) a2n = (2n)! (2n + 1)! Consequently, the series solution is given by ∞ ∞ (−1)n 2n (−1)n 2n+1 x + x u(x) = , (14.88) (2n)! (2n + 1)! n=0 n=0 that converges to the exact solution u(x) = cos x + sin x.

(14.89)

Example 14.12 Solve the nonlinear Volterra integro-diﬀerential equation by using the series solution method x 2 2 (1 + u2 (t))dt, (14.90) u (x) = 2 sec x(1 + 3 tan x) − tan x + 0

with initial conditions u(0) = u (0) = 0, u (0) = 1. Substituting u(x) by the series ∞ an xn , (14.91) u(x) = n=0

into both sides of equation (14.90), evaluating the integral at the right side, using a0 = a2 = 0, a1 = 1, and proceeding as before we ﬁnd 1 2 , a1 = 1, a3 = , a 5 = 3 15 (14.92) 17 , a2k = 0, a7 = k 0, 315 Consequently, the series solution is given by 2 17 7 1 x + ··· , (14.93) u(x) = x + x3 + x5 + 3 15 315 that converges to the exact solution u(x) = tan x.

(14.94)

Exercises 14.2.3 Solve the following nonlinear Volterra integro-diﬀerential equations by using the series solution method

440

14 Nonlinear Volterra Integro-Diﬀerential Equations x 1. u (x) = −x + tanh x(1 − sech x) + (1 − u2 (t))dt, u(0) = 1 0

1 3 2. u (x) = − x + cosh x + sinh(2x) + 2 4

x 0

(1 − u2 (t))dt, u(0) = 0

x 1 1 3. u (x) = − x + sin x + cos x + sin(2x) + (x − t)(1 − u2 (t))dt, u(0) = −1 2 4 0 x ex+t u2 (t)dt, u(0) = 1 4. u (x) = 1 − 2 cosh x + 0

5. u (x) = 1 − 2 sinh x +

x

0

ex+t u2 (t)dt, u(0) = 1, u (0) = −1

6. u (x) = 1 + ex (1 − 2x) − e2x + 7. u (x) = 2xex + e2x +

x 0

x

0

ex−t u2 (t)dt, u(0) = 2, u (0) = 1

ex−t (1 − u2 (t))dt, u(0) = 2, u (0) = u (0) = 1

1 1 8. u (x) = − x − 8 cos(2x) − sin(4x) + 2 8 u(0) = u (0) = 0, u (0) = 2

x 0

(1 − u2 (t))dt,

14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind The standard form of the nonlinear Volterra integro-diﬀerential equation [1– 3] of the ﬁrst kind is given by x x K1 (x, t)F (u(t))dt + K2 (x, t)u(i) (t)dt = f (x), (14.95) 0

0

where u(i) (x) is the i th derivative of u(x). For this equation, the kernels K1 (x, t) and K2 (x, t), and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x). For the determination of the exact solution, the initial conditions should be prescribed. The nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind (14.95) can be converted to a nonlinear Volterra integral equation of the second kind by integrating the second integral in (14.95) by parts. The nonlinear Volterra integro-diﬀerential equations of the ﬁrst kind will be handled in this section by the combined Laplace transform-Adomian decomposition method and by converting it to a nonlinear Volterra integro-diﬀerential equation of the second kind.

14.3.1 The Combined Laplace Transform-Adomian Decomposition Method The combined Laplace transform-Adomian decomposition method was used in the previous section for solving nonlinear Volterra integro-diﬀerential equa-

14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind

441

tions of the second kind. The analysis will be focused on equations where the kernels K1 (x, t) and K2 (x, t) of (14.95) are diﬀerence kernels. This means that each kernel depends on the diﬀerence (x − t). Recall that the Laplace transform of the convolution product (f1 ∗ f2 )(x) is given by x f1 (x − t)f2 (t)dt = F1 (s)F2 (s). (14.96) L{(f1 ∗ f2 )(x)} = L 0

Recall that L{u(n) (x)} = sn L{u(x)} − sn−1 u(0) − sn−2 u (0) − · · · u(n−1) (0). Taking Laplace transform of both sides of (14.95) gives L{K1 (x − t) ∗ F (u(x))} + L{K2 (x − t) ∗ u(i) (x)} = L{(f (x)}, so that K1 (s)L{F (u(x))} + K2 (s)L{u(i) (x)} = φ(s),

(14.97) (14.98) (14.99)

where φ(s) = L{f (x)}, K1 (s) = L{K1 (x)}, K2 (s) = L{K2 (x)}. Using (14.97) and solving for U (s) we ﬁnd φ(s) + K2 (s)Γ(s) − K1 (s)L{F (u(x))} U (s) = , si K2 (s) where Γ(s) = si−1 u(0) + si−2 u (0) + · · · + u(i−1) (0), U (s) = L{u(x)}.

(14.100)

(14.101)

(14.102)

The combined Laplace transform-Adomian decomposition method can be used eﬀectively in (14.101) provided that K1 (s) = 0. (14.103) lim s→∞ si K2 (s) To overcome the diﬃculty of the nonlinear term F (u(x)), we apply the Adomian decomposition method for handling (14.101). To achieve this goal, we ﬁrst represent the linear term u(x) at the left side by an inﬁnite series of components given by ∞ u(x) = un (x), (14.104) n=0

where the components un (x), n 0 will be recursively determined. However, the nonlinear term F (u(x)) at the right side of (14.101) will be represented by an inﬁnite series of the Adomian polynomials An in the form ∞ An (x), (14.105) F (u(x)) = n=0

where the Adomian polynomials An , n 0 are given by n 1 dn i An = F λ ui , n = 0, 1, 2, · · · . n! dλn i=0 λ=0

(14.106)

442

14 Nonlinear Volterra Integro-Diﬀerential Equations

Substituting (14.104) and (14.105) into (14.101) leads to + ∞ 1 1 1 1 un (x) = u(0) + 2 u (0) + · · · + i u(i−1) (0) + i L φ(s) s s s s K2 (s) n=0 + ∞ K1 (s) An (x) . L − i s K2 (s) n=0 (14.107) The Adomian decomposition method admits the use of the following recursive relation 1 1 1 1 U0 (s) = u(0) + 2 u (0) + · · · + i u(i−1) (0) + i φ(s), s s s s K2 (s) (14.108) K1 (s) L{uk+1 (x)} = − i L{Ak (x)}, k 0, s K2 (s) provided that K1 (s) = 0. (14.109) lim s→∞ si K2 (s) For example if K2 (x, t) = cosh(x − t), K1 (x, t) = x − t , and the equation includes u (x), then s2 = 1. (14.110) lim 2 s→∞ s − 1 In such a problem, the combined Laplace transform-Adomian decomposition method cannot be used. Instead another approach should be used to handle this case. Applying the inverse Laplace transform to the ﬁrst part of (14.108) gives u0 (x), that will deﬁne A0 . This in turn will lead to the complete determination of the components of uk , k 0. The analysis presented above will be illustrated by the following examples. Example 14.13 Solve the following nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind by the combined Laplace transform-Adomian decomposition method x x 7 1 1 (x − t)u2 (t)dt + (x − t)u (t)dt = + x2 − cos x + cos(2x), u(0) = 0. 8 4 8 0 0 (14.111) Taking Laplace transforms of both sides gives 1 1 s 1 7 s + 3− 2 + , (14.112) L{u2 }(s) + 2 (sU (s) − u(0)) = 2 2 s s 8s 2s s + 1 8(s + 4) where by using the given initial condition and solving for U (s) we obtain 1 s2 1 7 s2 + − L{u2 }(s). U (s) = + 2 − 2 (14.113) 8 2s s + 1 8(s2 + 4) s Substituting the series assumption for U (s) and the Adomian polynomials for u2 (x), and using the recursive relation (14.108) we obtain

14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind

443

1 s2 7 s2 + 2− 2 + , 8 2s s + 1 8(s2 + 4) (14.114) 1 L{uk+1 (x)} = − L{Ak (x)}, k 0. s Taking the inverse Laplace transform of both sides of the ﬁrst part of (14.114), and using the recursive relation (14.114) gives 1 7 5 x + ··· , u0 (x) = x + x3 − 3! 120 1 1 (14.115) u1 (x) = − x3 − x5 + · · · , 3 15 2 5 x + ··· . u2 (x) = 15 The series solution is therefore given by 1 1 (14.116) u(x) = x − x3 + x5 + · · · , 3! 5! that converges to the exact solution u(x) = sin x. (14.117) U0 (s) =

Example 14.14 Solve the following nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind by the combined Laplace transform-Adomian decomposition method x x 1 1 1 (x − t)u2 (t)dt + ex−t u (t)dt = − − x + xex + e2x , u(0) = 1. 4 2 4 0 0 (14.118) Taking Laplace transforms of both sides gives 1 1 1 1 s 1 (sU (s) − u(0)) = − − 2 + , L{u2 }(s) + + s2 s−1 4s 2s (s − 1)2 4(s − 2) (14.119) where by using the given initial condition and solving for U (s) we obtain 1 s−1 s 1 1 1 s−1 U (s) = + + − − 2+ − 3 L{u2 }(s). s s 4s 2s (s − 1)2 4(s − 2) s (14.120) Substituting the series assumption for U (s) and the Adomian polynomials for u2 (x), and using the recursive relation (14.108) we obtain 1 1 1 s−1 s 1 − − 2+ , U0 (s) = + + s s 4s 2s (s − 1)2 4(s − 2) (14.121) s−1 L{uk+1 (x)} = − 3 L{Ak (x)}, k 0. s Taking the inverse Laplace transform of both sides of the ﬁrst part of (14.121), and using the recursive relation (14.121) gives

444

14 Nonlinear Volterra Integro-Diﬀerential Equations

1 1 1 u0 (x) = 1 + x + x2 + x3 + x4 + x5 + · · · , 3 8 24 1 1 1 1 u1 (x) = − x2 − x3 − x4 − x5 + · · · , 2 6 6 12 1 4 1 5 x + x + ··· . u2 (x) = 12 20 The series solution is therefore given by 1 1 1 u(x) = 1 + x + x2 + x3 + x4 + x5 + · · · , 3! 4! 5! that converges to the exact solution u(x) = ex .

(14.122)

(14.123) (14.124)

Example 14.15 Solve the following nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind by the combined Laplace transform-Adomian decomposition method x x 15 1 1 cos2 (2x), (x − t)u2 (t)dt + (x − t)u (t)dt = − + x2 + cos(2x) − 16 4 16 0 0 (14.125) where u(0) = 1, u (0) = 0. Taking Laplace transforms of both sides and using the initial conditions we obtain 15 1 2 1 1 1 cos2 (2x) − 2 L{u2 }(s). (14.126) U (s) = +L − + x +cos(2x) − s 16 4 16 s Proceeding as before leads to 3 1 4 u0 (x) = 1 − x2 + x4 + x6 + · · · , 2 3 45 1 2 1 4 7 6 u1 (x) = − x + x − x + · · · , 2 4 72 (14.127) 1 4 1 6 u2 (x) = x + x + ··· , 12 15 1 6 u3 (x) = − x + · · · , 72 The series solution is therefore given by 1 1 1 (14.128) u(x) = 1 − (2x)2 + (2x)4 − (2x)6 + · · · , 2! 4! 6! that converges to the exact solution u(x) = cos(2x). (14.129) Example 14.16 Solve the following nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind by the combined Laplace transform-Adomian decomposition method x x 1 1 1 2 (x − t)u (t)dt + (x − t)u (t)dt = − − 3x + x2 + 3 sinh x + cosh2 x, 4 4 4 0 0 (14.130)

14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind

445

where u(0) = 1, u (0) = 1. Taking Laplace transforms of both sides and using the initial conditions we obtain 1 1 1 1 1 1 U (s) = + 2 + L{− − 3x + x2 + 3 sinh x + cosh2 x} − 2 L{u2 }(s). s s 4 4 4 s (14.131) Proceeding as before leads to 1 1 1 1 u0 (x) = 1 + x + x2 + x3 + x4 + x5 + · · · , 2 2 12 40 1 1 1 1 (14.132) u1 (x) = − x2 − x3 − x4 − x5 + · · · , 2 3 6 10 1 4 1 u2 (x) = x + x5 + · · · , 12 12 The series solution is therefore given by 1 1 u(x) = 1 + x + x3 + x5 + · · · , (14.133) 3! 5! that converges to the exact solution u(x) = 1 + sinh x.

(14.134)

Exercises 14.3.1 Solve the following nonlinear Volterra integro-diﬀerential equations of the ﬁrst kind by using the combined Laplace transform-Adomian decomposition method x x 1 1 1 1. (x − t)u2 (t)dt + (x − t)u (t)dt = − x + x2 + sin x − cos(2x), u(0) = 1 8 4 8 0 0 x x 1 1 1 2 2 (x − t)u (t)dt + (x − t)u (t)dt = − − x + x + sinh x + cosh(2x), 2. 8 4 8 0 0 u(0) = 1 x (x − t)u2 (t)dt + 3. 0

u(0) = 1 x ex−t u2 (t)dt + 4. 0 x

5. 0 x

6. 0

ex−t u2 (t)dt +

x 0

x

ex−t u (t)dt = −4 + 3ex + 5xex + e2x , u(0) = 3, u (0) = 1

0

1 1 1 x + x2 + sin x − cos x − sin(2x), 2 2 4

ex−t u (t)dt = −1 + 3xex + e2x , u(0) = 2

0 x

(x − t)u2 (t)dt +

(x − t)u (t)dt = 1 −

x 0

(x − t)u (t)dt = −

1 1 − 2x + x2 + sin(2x) 16 4

1 + cos2 x, u(0) = 0, u (0) = 2 16 x x 1 6 (x − t)u2 (t)dt + (x − t)u (t)dt = x2 − 7. x , u(0) = 0, u (0) = 0 36 0 0 x x 10 1 4 8. (x − t)u3 (t)dt + (x − t)u (t)dt = − − x + ex + e3x , 9 3 9 0 0 u(0) = 1, u (0) = 1

446

14 Nonlinear Volterra Integro-Diﬀerential Equations

14.3.2 Conversion to Nonlinear Volterra Equation of the Second Kind In this section we will convert the nonlinear Volterra integro-diﬀerential equation [1–3] of the ﬁrst kind x x K1 (x, t)F (u(t))dt + K2 (x, t)u(n) (t)dt = f (x), K2 (x, x) = 0, 0

0

(14.135) to a nonlinear Volterra integral equation of the second kind or nonlinear Volterra integro-diﬀerential equation of the second kind. Without loss of generality, we will study the cases of the ﬁrst and second order derivatives of the form x

0

x

K1 (x, t)F (u(t))dt +

and x 0

0

K1 (x, t)F (u(t))dt +

x 0

K2 (x, t)u (t)dt = f (x), K2 (x, x) = 0, (14.136)

K2 (x, t)u (t)dt = f (x), K2 (x, x) = 0. (14.137)

However, equations of higher order can be handled in a similar manner. Integrating the second integral in (14.136) by parts gives the nonlinear Volterra integral equation x x ∂K2 (x, t) u(t)dt K1 (x, t)F (u(t))dt + K2 (x, x)u(x) − K2 (x, 0)u(0) − ∂t 0 0 = f (x), (14.138) or equivalently x K2 (x, 0) 1 f (x) ∂(K2 (x, t)) + u(0) + u(t)dt u(x) = K2 (x, x) K2 (x, x) K2 (x, x) 0 ∂t (14.139) x 1 − K1 (x, t)F (u(t))dt, K2 (x, x) = 0. K2 (x, x) 0 Equation (14.139) is the nonlinear Volterra integral equation of the second kind that was handled in this chapter by distinct methods. In a like manner, we integrate the second integral in (14.137) by parts to obtain the nonlinear Volterra integro-diﬀerential equation of the second kind x K1 (x, t)F (u(t))dt + K2 (x, x)u (x) − K2 (x, 0)u (0) 0 (14.140) x ∂K2 (x, t) u (t)dt = f (x), K2 (x, x) = 0. − ∂t 0 or equivalently

14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind

447

x ∂(K2 (x, t)) K2 (x, 0) f (x) 1 + u (0) + u (t)dt K2 (x, x) K2 (x, x) K2 (x, x) 0 ∂t (14.141) x 1 K1 (x, t)F (u(t))dt, K2 (x, x) = 0. − K2 (x, x) 0 It is important to notice that if the nonlinear Volterra integral equation of the ﬁrst kind contains the ﬁrst derivative of u(x), then the conversion process will give a nonlinear Volterra integral equation of the second kind as shown by (14.139). However, if the nonlinear Volterra integral equation of the ﬁrst kind contains u(i) (x), i 2, then integrating the second integral once will give a nonlinear Volterra integro-diﬀerential equation of the second kind as shown by (14.141). Both types of equations were examined before. In what follows, the ﬁrst two illustrative examples will be handled by the modiﬁed decomposition method and the other two examples will be handled by using the variational iteration method. Other methods can be used as well.

u (x) =

Example 14.17 Convert the nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind x x 1 1 1 (x−t)u2 (t)dt+ ex−t u (t)dt = xex + e2x − − x, u(0) = 1, (14.142) 4 4 2 0 0 to a nonlinear Volterra integral equation of the second kind and solve it. Integrating the second integral by parts, using the initial condition, and solving for u(x) we ﬁnd x x 1 1 1 u(x) = ex +xex + e2x − − x− (x−t)u2 (t) dt− ex−tu(t) dt. (14.143) 4 4 2 0 0 We select the modiﬁed decomposition method, therefore we use the recurrence relation approximations u0 (x) = ex + xex , x x 1 2x 1 1 2 u1 (x) = e − − x − (x − t)u0 (t)dt − ex−tu0 (t)dt, (14.144) 4 4 2 0 0 = −xex + · · · . Canceling the noise term xex from u0 (x) gives the exact solution u(x) = ex .

(14.145)

Example 14.18 Convert the nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind x x 1 1 2 u (t) dt+ (x−t+1) u(t) dt = sin x+cos x+ sin(2x)−1− x, u(0) = 1, 4 2 0 0 (14.146) to a nonlinear Volterra integral equation of the second kind and solve it. Integrating the second integral by parts, using the initial condition, and solving for u(x) we ﬁnd

448

14 Nonlinear Volterra Integro-Diﬀerential Equations

x x 1 1 2 u(x) = sin x + cos x + sin(2x) + x − u (t) dt − u(t) dt. (14.147) 4 2 0 0 We again select the modiﬁed decomposition method, therefore we use the recurrence relation approximations u0 (x) = sin x + cos x, x x 1 1 2 u1 (x) = sin(2x) + x − u0 (t)dt − u0 (t)dt, (14.148) 4 2 0 0 = − sin x + · · · . Canceling the noise term sin x from u0 (x) gives the exact solution u(x) = cos x. (14.149) Example 14.19 Convert the nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind x x 1 1 u2 (t)dt+ (x−t+1)u (t)dt = sin x−cos x− sin(2x)+1+ x, (14.150) 4 2 0 0 with u(0) = u (0) = 1, to a nonlinear Volterra integro-diﬀerential equation of the second kind and solve it. Integrating the second integral by parts, using the initial conditions, and solving for u(x) we ﬁnd x x 3 1 u2 (t)dt− u (t)dt. (14.151) u (x) = 2 + x+ sin x− cos x− sin(2x)− 2 4 0 0 This equation will be solved by using the variational iteration method. The correction functional for this equation is given by un+1 (x) = un (x) x t 1 3 un (t) − 2 − t − sin t + cos t + sin(2t) + − (un (r) + u2n (r))dr dt. 2 4 0 0 (14.152) We can use the initial conditions to select u0 (x) = 1+x. Using this selection into the correction functional gives the following successive approximations u0 (x) = 1 + x, 1 1 1 u1 (x) = 1 + x − x3 − x4 − x5 + · · · , 3! 4! 5! (14.153) 1 1 u2 (x) = 1 + x − x3 + x5 + · · · , 3! 60 1 1 u3 (x) = 1 + x − x3 + x5 + · · · , 3! 5! and so on. The VIM gives the exact solution by u(x) = 1 + sin x.

(14.154)

Example 14.20 Convert the nonlinear Volterra integro-diﬀerential equation of the ﬁrst kind

14.3 Nonlinear Volterra Integro-Diﬀerential Equations of the First Kind

x

449

x

3 1 ex−t u (t) dt = − + ex (2 − x) − e2x , (14.155) 2 2 0 0 with u(0) = 1, u (0) = 2, to a nonlinear Volterra integro-diﬀerential equation of the second kind and solve it. Integrating the second integral by parts, using the initial equation, and solving for u(x) we ﬁnd x x 3 1 (t2 −u2 (t)) dt− ex−t u (t) dt. (14.156) u (x) = − +ex (4−x)− e2x − 2 2 0 0 This equation will be solved by using the variational iteration method. The correction functional for this equation is given by un+1 (x) = un (x) t x 3 1 (r2 − u2n (r) + et−r un (r))dr dt. − un(t) + − et (4 − t) + e2t + 2 2 0 0 (14.157) Using the initial conditions to select u0 (x) = u(0) + xu (0) = 1 + 2x, we obtain the following successive approximations u0 (x) = 1 + 2x, 1 1 1 u1 (x) = 1 + 2x + x2 + x3 + x4 + · · · , 2! 3! 4! 1 2 1 3 1 u2 (x) = 1 + 2x + x + x + x5 + · · · , (14.158) 2! 3! 60 1 2 1 3 1 4 u3 (x) = 1 + 2x + x + x + x + · · · , 2! 3! 4! 1 2 1 3 1 1 u4 (x) = 1 + 2x + x + x + x4 + x5 + · · · , 2! 3! 4! 5! and so on. The VIM gives the exact solution by (t2 − u2 (t)) dt +

u(x) = x + ex .

(14.159)

Exercises 14.3.2 In Exercises 1–4, convert nonlinear Volterra integro-diﬀerential equations of the ﬁrst kind to nonlinear Volterra integral equations of the second kind and solve the resulting equations by any method x x 1 (cosh2 t − u2 (t))dt + cosh(x − t)u (t)dt = −x − 2 sinh x + x sinh x, u(0) = 2 1. 2 0 0 x x 1 (cos2 t − u2 (t))dt + cos(x − t)u (t)dt = −x − 2 sin x − x sin x, u(0) = 2 2. 2 0 0 x x 5 1 (t2 − u2 (t))dt + ex−t u (t)dt = − + ex (3 − x) − e2x , u(0) = 1 3. 2 2 0 0 x x 1 1 2 (1 − u (t))dt + cos(x − t)u (t)dt = sin x(2 sin x + x + 1) + x cos x, 4. 2 2 0 0 u(0) = −1

450

14 Nonlinear Volterra Integro-Diﬀerential Equations

In Exercises 5–8, convert nonlinear Volterra integro-diﬀerential equations of the ﬁrst kind to a nonlinear Volterra integro-diﬀerential equations of the second kind and solve the resulting equations by any method x x 1 1 5. (u2 (t) − 1)dt + cos(x − t)u (t)dt = sin x(2 sin x − x − 1) − x cos x, 2 2 0 0 u(0) = 1, u (0) = 1 x (sin2 t − u2 (t))dt + 6. 0

x 0

1 1 cos(x − t)u (t)dt = − x3 − (x + 4) sin x 3 2

+2x cos x, u(0) = 0, u (0) = 2 x x ex−4t u2 (t)dt + ex−2t u (t)dt = 5xex , u(0) = 1, u (0) = 2 7.

0 x

8. 0

0

2

(x − t)u (t)dt +

x 0

1 cosh(x − t)u (t)dt = − (1 + x2 ) 4

1 1 + x sinh x + cosh2 x, u(0) = 0, u (0) = 1 2 4

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations In this section, we will study systems of nonlinear Volterra integro-diﬀerential equations of the second kind given by x % 1 (x, t)F%1 (v(t)) dt, u(i) (x) = f1 (x) + K1 (x, t)F1 (u(t)) + K 0 (14.160) x (i) % % K2 (x, t)F2 (u(t)) + K2 (x, t)F2 (v(t)) dt. v (x) = f2 (x) + 0

The nonlinear functions Fi , F%i , i = 1, 2 of the unknown functions u(x), v(x) occur inside the integral sign whereas the derivatives of u(x), v(x) appear % i (x, t), and the mostly outside the integral sign. The kernels Ki (x, t) and K functions fi (x), i = 1, 2 are given real-valued functions. To determine the exact solutions for the system (14.160), the initial conditions u(j−1) (0) and v (j−1) (0), 1 j i should be prescribed. There is a variety of numerical and analytical methods that are usually used for solving the systems of nonlinear Volterra integro-diﬀerential equations (14.160). However, in this section, we will concern ourselves with two methods, namely the variational iteration method and the combined Laplace transform-Adomian method.

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

451

14.4.1 The Variational Iteration Method The variational iteration method (VIM) was used to handle Volterra integral equations and Volterra integro-diﬀerential equations. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the so called Adomian polynomials that we need to express nonlinear terms. The correction functionals for the Volterra system of integro-diﬀerential equations (14.160) are given by x t un+1 (x) = un (x) + λ(t) u(i) (t) − f (t) − γ (t, r)dr dt, 1 1 n 0 0 (14.161) x t vn+1 (x) = vn (x) + where

0

λ(t) vn(i) (t) − f2 (t) −

0

γ2 (t, r)dr dt.

% 1 (t, r)F%1 (˜ γ1 (t, r) = K1 (t, r)F1 (˜ un (r)) + K vn (r)), % 2 (t, r)F%2 (˜ γ2 (t, r) = K2 (t, r)F2 (˜ un (r)) + K vn (r)).

(14.162)

The variational iteration method is used by applying two essential steps. It is required ﬁrst to determine the Lagrange multiplier λ that can be identiﬁed optimally via integration by parts and by using a restricted variation. Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), n 0 and vn+1 (x), n 0 of the solutions u(x) and v(x). The zeroth approximations u0 (x) and v0 (x) can be any selective functions. However, using the initial conditions are preferably used for the selective zeroth approximations u0 and v0 (x) as will be seen later. Consequently, the solutions are given by (14.163) u(x) = lim un (x), v(x) = lim vn (x). n→∞

n→∞

The VIM will be illustrated by studying the following systems of nonlinear Volterra integro-diﬀerential equations of the second kind. Example 14.21 Use the VIM to solve the system of nonlinear Volterra integro-diﬀerential equations x 1 2 1 4 u (x) = 1 − x + x − x + ((x − t)u2 (t) + v 2 (t))dt, 2 12 0 (14.164) x 3 2 1 4 2 2 v (x) = −1 − x − x − x + (u (t) + (x − t)v (t))dt, 2 12 0 where u(0) = 1, v(0) = 1. The correction functionals for this system are

452

14 Nonlinear Volterra Integro-Diﬀerential Equations

x 1 2 1 4 un+1 (x) = un (x) − un (t) − 1 + t − t + t − I1 (t) dt, 2 12 0 (14.165) x 3 1 vn (t) + 1 + t + t2 + t4 − I2 (t) dt, vn+1 (x) = vn (x) − 2 12 0 where t

I1 (t) =

0

I2 (t) =

0

t

((t − r)u2n (r) + vn2 (r))dr,

(14.166) (u2n (r)

+ (t −

r)vn2 (r))dr,

and λ = −1 for ﬁrst order integro-diﬀerential equation. Selecting u0 (x) = u(0) = 1 and v0 (x) = v(0) = 1 gives the successive approximations u0 (x) = 1,

v0 (x) = 1, 1 1 1 u1 (x) = 1 + x + x3 − x4 − x5 , 3 6 60 1 1 v1 (x) = 1 − x − x3 − x5 , 3 60 1 3 1 3 1 4 1 4 1 u2 (x) = 1 + x + x − x + x − x − x5 + · · · , 3 3 6 6 30 1 1 4 1 4 1 3 1 3 v2 (x) = 1 − x + x − x + x − x + x5 + · · · , 3 3 6 6 30 1 5 1 5 u3 (x) = 1 + x + x − x + ··· , 30 30 1 5 1 5 x − x + ··· , v3 (x) = 1 − x + 30 30 and so on. It is obvious that the noise terms appear in each approximation, hence the exact solutions are given by (u(x), v(x)) = (1 + x, 1 − x). (14.167) Example 14.22 Use the VIM to solve the system of Volterra integro-diﬀerential equations x 1 1 3 u (x) = ex − e2x − x4 + x + + ((x − t)u2 (t) + (x − t)v 2 (t))dt, 2 6 2 0 x ((x − t)u2 (t) − (x − t)v 2 (t))dt, v (x) = 7ex − 4xex − 4x − 7 + 0

(14.168) where u(0) = 1, v(0) = −1. The correction functionals for this system are given by

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

453

1 2t 1 4 3 t un+1 (x) = un (x) − un(t) − e + e + t − t − − I1 (t) dt, 2 6 2 0 x

vn+1 (x) = vn (x) − vn (t) − 7et + 4tet + 4t + 7 − I2 (t) dt, x

0

where

(14.169)

I1 (t) = I2 (t) =

t 0

((t − r)u2n (r) + (t − r)vn2 (r))dr, (14.170)

t 0

((t −

r)u2n (r)

− (t −

r)vn2 (r))dr,

and λ = −1 for ﬁrst-order integro-diﬀerential equations. We can use the initial conditions to select u0 (x) = u(0) = 1 and v0 (x) = v(0) = −1. Using this selection into the correction functionals gives the following successive approximations v0 (x) = −1, 1 1 1 11 5 x + ··· , u1 (x) = 1 + 2x + x2 + x3 − x4 − 2! 3! 8 120 1 1 5 3 v1 (x) = −1 − x2 − x3 − x4 − x5 + · · · , 2! 3! 24 40 1 1 1 1 u2 (x) = 1 + 2x + x2 + x3 + x4 + x5 + · · · , 2! 3! 4! 5! 1 1 1 1 v2 (x) = −(1 + x2 + x3 + x4 + x5 + · · · ), 2! 3! 4! 5! and so on. The exact solutions are therefore given by (u(x), v(x)) = (x + ex , x − ex ). u0 (x) = 1,

(14.171)

Example 14.23 Use the VIM to solve the system of nonlinear Volterra integro-diﬀerential equations 1 1 1 u (x) = cosh x − sinh2 x − x4 − x2 2 6 2 x + ((x − t)u2 (t) + (x − t)v 2 (t))dt, u(0) = 1, u (0) = 1, 0

v (x) = −(1 + 4x) cosh x + 8 sinh x − 4x x ((x − t)u2 (t) − (x − t)v 2 (t))dt, v(0) = −1, v (0) = 1. + 0

The correction functionals for this system are given by

(14.172)

454

14 Nonlinear Volterra Integro-Diﬀerential Equations

un+1 (x) = un (x) x t 1 1 1 (t − x) un (t) − cosh t + sinh2 t + t4 + t2 − + I1 (t, r)dr dt, 2 6 2 0 0 (14.173) vn+1 (x) = vn (x) t x I2 (t, r)dr dt, (t − x) vn (t) + (1 + 4t) cosh t − 8 sinh t + 4t − + 0

0

where

I1 (t, r) = (t − r)u2 (r) + (t − r)v2 (r), I2 (t, r) = (t − r)u2 (r) − (t − r)v2 (r).

(14.174)

and λ = t − x for second-order integro-diﬀerential equation. Using u0 (x) = 1+x and v0 (x) = −1+x gives the successive approximations v0 (x) = −1 + x, 1 1 1 6 x + ··· , 1 + x + x2 + x4 − 2! 4! 240 1 1 1 6 x + ··· , −1 + x − x2 − x4 − 2! 4! 720 1 1 1 x + 1 + x2 + x4 + x6 + · · · , 2! 4! 6! 1 1 1 x − 1 + x2 + x4 + x6 + · · · , 2! 4! 6!

u0 (x) = 1 + x, u1 (x) = v1 (x) = u2 (x) = v2 (x) =

and so on. The exact solutions are therefore given by (u(x), v(x)) = (x + cosh x, x − cosh x).

(14.175)

Example 14.24 Use the variational iteration method to solve the following system e4x 3 e2x (x − 1) + (3x − 1) + (x + 1) u (x) = ex + 2 4 4 x ((x − 2t)u2 (t) + (x − 4t)v 2 (t))dt, u(0) = 1, u (0) = 1, + 0

e6x 5 e4x (3x − 1) + (5x − 1) + (x + 1) v (x) = 4e2x + 4 6 12 x + ((x − 4t)v 2 (t) + (x − 6t)w2 (t))dt, v(0) = 1, v (0) = 2, 0

e6x 2 e2x (x − 1) + (5x − 1) + (x + 1) w (x) = 9e + 2 6 3 x 2 2 ((x − 6t)w (t) + (x − 2t)u (t))dt, w(0) = 1, w (0) = 3. +

3x

0

(14.176) We select the zeroth approximations as u0 (x) = 1 + x, v0 (x) = 1 + 2x and w0 (x) = 1 + 3x. Proceeding as before, we obtain

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

u0 (x) = 1 + x,

v0 (x) = 1 + 2x,

u1 (x) = 1 + x +

1 2 1 1 1 x + x3 + x 4 + x5 + · · · , 2! 3! 4! 5!

v1 (x) = 1 + 2x +

455

w0 (x) = 1 + 3x,

1 1 1 1 (2x)2 + (2x)3 + (2x)4 + (2x)5 + · · · , 2! 3! 4! 5!

1 1 1 1 (3x)2 + (3x)3 + (3x)4 + (3x)5 + · · · , 2! 3! 4! 5! The exact solutions are therefore given by (u(x), v(x), w(x)) = (ex , e2x , e3x ). (14.177) w1 (x) = 1 + 3x +

Exercises 14.4.1 Use the variational iteration method to solve the following systems of nonlinear Volterra integro-diﬀerential equations x ⎧ 1 4 2 6 ⎪ ⎪ (x) = 2x + + + (x − 2t)(u2 (t) + v(t))dt x x u ⎪ ⎪ 6 15 ⎪ 0 ⎪ ⎨ x 1 4 2 6 1. v x x (x) = −2x − + + (x − 2t)(u(t) + v 2 (t))dt ⎪ ⎪ ⎪ 6 15 0 ⎪ ⎪ ⎪ ⎩ u(0) = 1, v(0) = 1 x ⎧ 1 3 8 ⎪ ⎪ x + (x − 2t)(u2 (t) + v 2 (t)) dt u (x) = 1 + 3x2 + x4 + ⎪ ⎪ 3 14 ⎪ 0 ⎪ ⎨ x 8 6 2. 2 x (x) = 1 − 3x + + (x − 2t)(u2 (t) − v 2 (t))dt v ⎪ ⎪ ⎪ 15 0 ⎪ ⎪ ⎪ ⎩ u(0) = 0, v(0) = 0 x ⎧ ⎪ x 2x ⎪ (x) = e − 2e + 2 + ex−t (u2 (t) + v 2 (t))dt u ⎪ ⎪ ⎪ 0 ⎪ ⎨ x 3. x x (x) = −e − 4xe + ex−t (u2 (t) − v 2 (t))dt v ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ u(0) = 2 v(0) = 0 x ⎧ 7 2 ⎪ ⎪ u (x) = 1 + sin x − sin(2x) − 4x + cos(x − t)(u2 (t) + v 2 (t))dt ⎪ ⎪ 3 3 ⎪ 0 ⎪ ⎨ x 4. sin(x − t)(u2 (t) − v 2 (t))dt v (x) = 1 + (2 − x2 ) sin x − x cos x + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ u(0) = 1, v(0) = −1 x ⎧ 1 ⎪ ⎪ u (x) = 2 − 2x2 − 2x3 − x4 + (x − t + 1)(u2 (t) − v 2 (t))dt ⎪ ⎪ 3 ⎪ 0 ⎪ ⎨ x 2 2 5. v (x) = −2 + x3 + x4 + (x − 2t)(u2 (t) − v 2 (t))dt ⎪ ⎪ ⎪ 3 3 0 ⎪ ⎪ ⎪ ⎩ u(0) = u (0) = 1, v(0) = 1, v (0) = −1

456 14 Nonlinear Volterra Integro-Diﬀerential Equations x ⎧ 5 2 ⎪ ⎪ (x) = cos(x − t)(u2 (t) + v 2 (t))dt sin x + sin(2x) − 4x + u ⎪ ⎪ 3 3 ⎪ 0 ⎪ ⎨ x 6. 2 (x) = (1 − x) sin x + x cos x + sin(x − t)(u2 (t) − v 2 (t))dt v ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ u(0) = 0, u (0) = 2, v(0) = v (0) = 0 x ⎧ ⎪ (x) = (1 − 2x)ex + ⎪ ex−2t (u2 (t) + v(t))dt u ⎪ ⎪ ⎪ 0 ⎪ ⎪ x ⎪ ⎪

x−4t 2 ⎪ ⎪ v (x) = 4e2x − 2xex + ⎨ v (t) + ex−3t w(t) dt e 0 7. ⎪ x ⎪

x−6t 2 ⎪ ⎪ 3x x ⎪ w (t) + ex−t u(t) dt w (x) = 9e − 2xe + e ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ u(0) = u (0) = 1, v(0) = 1, v (0) = 2, w(0) = 1, w (0) = 3 ⎧ 2x e 3 ⎪ ⎪ u (x) = ex + (x − 1) + e4x (3x − 1) + (x + 1) ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ x ⎪ ⎪

⎪ 2 2 ⎪ (x − 2t)u (t) + (x − 4t)v (t) dt + ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ 6x ⎪ ⎪ v (x) = 8e2x + e4x (3x − 1) + 3e (5x − 1) + 5 (x + 1) ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎨ x

8. (x − 4t)v 2 (t) + (x − 6t)w 2 (t) dt + ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ e2x ⎪ 3e6x ⎪ ⎪ w (x) = 27e3x + (x − 1) + (5x − 1) + 2(x + 1) ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ x ⎪ ⎪

⎪ ⎪ ⎪ (x − 6t)w 2 (t) + (x − 2t)u2 (t) dt + ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎩ u(0) = u (0) = 1, v(0) = 1, v (0) = 4, w(0) = 1, w (0) = 9

14.4.2 The Combined Laplace Transform-Adomian Decomposition Method The combined Laplace transform-Adomian decomposition method was used in this chapter to handle nonlinear Volterra integral equations where it worked eﬀectively. We will use the combined Laplace transform-Adomian decomposition method to study systems of nonlinear Volterra integro-diﬀerential equations of the second kind x % 1 (x, t)F%1 (v(t)))dt, u(i) (x) = f1 (x) + (K1 (x, t)F1 (u(t)) + K 0 (14.178) x % 2 (x, t)F%2 (v(t)))dt. (K2 (x, t)F2 (u(t)) + K v (i) (x) = f2 (x) + 0

The nonlinear functions Fi , F%i , i = 1, 2 of the unknown functions u(x), v(x) occur inside the integral sign whereas the derivatives of u(x) and v(x) appear

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

457

% i (x, t), and the mostly outside the integral sign. The kernels Ki (x, t) and K functions fi (x), i = 1, 2 are given real-valued functions. % i (x, t) as diﬀerence kernels We will consider the kernels Ki (x, t) and K where each kernel depends on the diﬀerence x − t. To determine the exact solutions for the system (14.178), the initial conditions u(j−1) (0) and v (j−1) (0), 1 j i should be prescribed. To use the combined-Laplace transform-Adomian decomposition method, recall that the Laplace transforms of the derivatives of u(x) are given by L{u(i) (x)} = si L{u(x)} − si−1 u(0) − si−2 u (0) − · · · − u(i−1) (0). (14.179) Applying the Laplace transforms to both sides of (14.178) gives si L{u(x)} − si−1 u(0) − si−2 u (0) − · · · − u(i−1) (0) % 1 (x − t) ∗ F%1 (v(t))}, = L{f1 (x)} + L{K1 (x − t) ∗ F1 (u(t)) + K si L{v(x)} − si−1 v(0) − si−2 v (0) − · · · − v(i−1) (0) % 2 (x − t) ∗ F%2 (v(t))}, = L{f2 (x)} + L{K2 (x − t) ∗ F2 (u(t)) + K

(14.180)

or equivalently 1 1 1 1 L{u(x)} = u(0) + 2 u (0) + · · · + i u(i−1) (0) + i L{f1 (x)} s s s s 1 % 1 (x − t) ∗ F%1 (v(t))}, + i L{K1 (x − t) ∗ F1 (u(t)) + K s (14.181) 1 1 1 1 L{v(x)} = v(0) + 2 v (0) + · · · + i v (i−1) (0) + i L{f2 (x)} s s s s 1 % 2 (x − t) ∗ F%2 (v(t))}. + i L{K2 (x − t) ∗ F2 (u(t)) + K s To overcome the diﬃculty of the nonlinear terms Fi (u(x)), i = 1, 2, we apply the Adomian decomposition method for handling (14.181). To achieve this goal, we ﬁrst represent the linear terms u(x) and v(x) at the left side by an inﬁnite series of components given by ∞ ∞ u(x) = un (x), v(x) = vn (x), (14.182) n=0

n=0

and the nonlinear terms Fi (u(x)) at the right side of (14.181) by n ∞ 1 dn i F F (u(x)) = An (x), An = λ ui , n = 0, 1, 2, · · · , n! dλn n=0 i=0 λ=0 (14.183) where the Adomian polynomials An , n 0 can be obtained for all forms of nonlinearity. Substituting (14.182) and (14.183) into (14.181) leads to + ∞ 1 1 1 1 un (x) = u(0) + 2 u (0) + · · · + i u(i−1) (0) + i L{f1 (x)} L s s s s n=0 ∞ + 1 + i L{K1 (x − t)}L An (x) s n=0

458

14 Nonlinear Volterra Integro-Diﬀerential Equations

∞

∞ + 1 % 1 (x − t)}L %n (x) , + i L{K A s n=0

+

1 1 1 1 v(0) + 2 v (0) + · · · + i v(i−1) (0) + i L{f2 (x)} s s s s n=0 ∞ + 1 + i L{K2 (x − t)}L Bn (x) s n=0 ∞ + 1 % ˜ + i L{K2 (x − t)}L Bn (x) . s n=0 (14.184) The Adomian decomposition method admits the use of the following recursive relations 1 1 1 1 L{u0 (x)} = u(0) + 2 u (0) + · · · + i u(i−1) (0) + i L{f1 (x)}, s s s s 1 1 % 1 (x − t)}L{A %k (x)}, L{uk+1 (x)} = i L{K1 (x − t)}L{Ak (x)} + i L{K s s and 1 1 1 1 L{v0 (x)} = v(0) + 2 v (0) + · · · + i v (i−1) (0) + i L{f2 (x)}, s s s s 1 1 % 2 (x − t)}L{B %k (x)}, L{vk+1 (x)} = i L{K2 (x − t)}L{Bk (x)} + i L{K s s (14.185) for k 0. The necessary conditions presented in Chapter 1 for Laplace transform method concerning the limit as s → ∞, should be satisﬁed here for a successful use of this method. Applying the inverse Laplace transform to the ˜0 . ﬁrst part of (14.185) gives u0 (x) and v0 (x), that will deﬁne A0 , A˜0 , B0 , B This in turn will lead to the complete determination of the components of uk+1 , vk+1 (x, k 0 upon using the second part of (14.185). The combined Laplace transform Adomian-decomposition method for solving systems of nonlinear Volterra integro-diﬀerential equations of the second kind will be illustrated by studying the following examples. L

vn (x)

=

Example 14.25 Solve the system of nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method x

ex−t u2 (t) + v 2 (t) dt, u(0) = 2, u (x) = ex − 2e2x + 2 + 0 (14.186) x 2

x−t 2 x x u (t) − v (t) dt, v(0) = 0. v (x) = −e − 4xe + 2 + e 0

Notice that the four kernels are K(x − t) = ex−t . Taking Laplace transforms of both sides of (14.186) gives

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

459

L{u (x)} = L{ex − 2e2x + 2} + L{ex−t ∗ (u2 (x) + v2 (x))}, (14.187) L{v (x)} = L{−ex − 4xex } + L{ex−t ∗ (u2 (x) − v2 (x))}, so that 1 2 2 1 sU (s) − u(0) = − + + L{u2 (x) + v 2 (x)}, s−1 s−2 s s−1 (14.188) 1 1 4 2 2 + (x) − v (x)}, sV (s) − v(0) = − − L{u s − 1 (s − 1)2 s−1 or equivalently 1 2 2 2 1 − + L{u2 (x) + v 2 (x)}, U (s) = + + s s(s − 1) s(s − 2) s2 s(s − 1) 4 1 1 − L{u2 (x) − v2 (x)}. + V (s) = − s(s − 1) s(s − 1)2 s(s − 1) (14.189) Substituting the series assumption for U (s) and V (s), and the Adomian polynomials for u2 (x) and v2 (x) as given above in (14.182) and (14.183) respectively, and using the recursive relation (14.185) we obtain 1 2 2 2 + 2− , U0 (s) = + s s(s − 1) s s(s − 2) 1 Uk+1 (s) = L{Ak (x) + Bk (x)}, s(s − 1) and 1 4 , V0 (s) = − − s(s − 1) s(s − 1)2 (14.190) 1 Vk+1 (s) = L{Ak (x) − Bk (x)}. s(s − 1) Recall that the Adomian polynomials for u2 (x) and v 2 (x) are given by A0 (x) = u20 , B0 (x) = v02 , A1 (x) = 2u0 u1 , B1 (x) = 2v0 v1 , (14.191) B2 (x) = 2v0 v2 + v12 , A2 (x) = 2u0 u2 + u21 , A3 (x) = 2u0 u3 + 2u1 u2 , B3 (x) = 2v0 v3 + 2v1 v2 . Taking the inverse Laplace transform of both sides of (14.190), and using the recursive relation (14.190) gives 3 7 5 31 5 x + ··· , u0 (x) = 2 + x − x2 − x3 − x4 − 2 6 8 120 4 2 (14.192) u1 (x) = 2x2 + x3 − x5 + · · · , 3 15 2 2 u2 (x) = x4 + x5 + · · · , 3 5 and

460

14 Nonlinear Volterra Integro-Diﬀerential Equations

5 3 13 17 5 x + ··· , v0 (x) = −x − x2 − x3 − x4 − 2 2 24 120 4 1 2 v1 (x) = 2x2 + x3 − x4 − x5 , · · · , 3 6 3 2 4 4 5 v2 (x) = x + x + · · · . 3 5 Using (14.182) gives the series solutions by 1 3 1 4 1 5 1 2 u(x) = 1 + 1 + x + x + x + x + x + · · · , 2! 3! 4! 5! 1 2 1 3 1 4 1 v(x) = 1 − 1 + x + x + x + x + x5 + · · · . 2! 3! 4! 5! Consequently, the exact solutions are given by (u(x), v(x)) = (1 + ex , 1 − ex ).

(14.193)

(14.194)

(14.195)

Example 14.26 Solve the system of nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method x 7 2 cos(x − t)(u2 (t) + v 2 (t))dt, u (x) = 1 + sin x − sin(2x) − 4x + 3 3 0 x v (x) = 1 − 7 sin x + 4x cos x + 4x + (x − t)(u2 (t) − v 2 (t))dt, 0

u(0) = 1, v(0) = −1. (14.196) Taking Laplace transforms of both sides of (14.196) gives 7 2 L{u (x)} = L 1 + sin x − sin(2x) − 4x 3 3 +L{cos(x − t) ∗ (u2 (x) + v2 (x))}, (14.197) L{v (x)} = L{1 − 7 sin x + 4x cos x + 4x} +L{(x − t) ∗ (u2 (x) − v2 (x))}, so that 7 4 4 1 s − − L{u2 (x) + v 2 (x)}, sU (s) − u(0) = + + 2 s 3(s2 + 1) 3(s2 + 4) s2 s +1 7 4(s2 − 1) 1 4 1 + 2 sV (s) − v(0) = − 2 + 2 + 2 L{u2 (x) − v 2 (x)}, s s + 1 (s + 1)2 s s (14.198) or equivalently 1 4 4 7 1 1 − − L{u2 (x) + v 2 (x)}, + 2 U (s) = + 2 + s s 3s(s2 + 1) 3s(s2 + 4) s3 s +1 1 4s(s2 − 1) 7 4 1 1 + + 3 + 3 L{u2 (x) − v 2 (x)}. V (s) = − + 2 − s s s(s2 + 1) (s2 + 1)2 s s (14.199)

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

461

Substituting the series assumption for U (s) and V (s), and the Adomian polynomials for u2 (x) and v2 (x) as given above in (14.182) and (14.183) respectively, and using the recursive relation (14.185) we obtain 1 4 4 1 7 U0 (s) = + 2 + − − , s s 3s(s2 + 1) 3s(s2 + 4) s3 1 Uk+1 (s) = 2 L{Ak (x) + Bk }, s +1 and 1 7 4 1 4s(s2 − 1) + 3, V0 (s) = − + 2 − + 2 2 s s s(s + 1) (s + 1)2 s (14.200) 1 Vk+1 (s) = 3 L{Ak (x) − Bk }. s Using the Adomian polynomials, and taking the inverse Laplace transform of both sides of the ﬁrst part of (14.190), and using the recursive relation (14.200) gives 3 1 19 6 x + ··· , u0 (x) = 1 + x − x2 + x4 − 2 8 720 1 1 41 6 x + ··· , u1 (x) = x2 − x4 − x5 + 4 10 360 (14.201) 1 4 1 5 2 6 u2 (x) = x + x − x + · · · , 6 10 15 2 6 x + ··· , u3 (x) = 45 and 1 5 13 6 x + ··· , v0 (x) = −1 + x + x2 − x4 + 2 24 720 1 1 1 (14.202) v1 (x) = x4 − x5 − x6 , · · · , 6 30 30 1 5 1 v2 (x) = x + x6 . 30 60 This in turn gives the series solutions by 1 2 1 4 1 6 u(x) = x + 1 − x + x − x + · · · , 2! 4! 6! (14.203) 1 4 1 1 2 v(x) = x − 1 − x + x − x6 + · · · . 2! 4! 6! Consequently, the exact solutions are given by (u(x), v(x)) = (x + cos x, x − cos x). (14.204) Example 14.27 Solve the system of nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method

462

14 Nonlinear Volterra Integro-Diﬀerential Equations

x

(u2 (t) + v 2 (t))dt, u(0) = 1, v(0) = 1, x v (x) = − cos x − sin x + 2 cos2 x − 2 + (u2 (t) − v 2 (t))dt.

u (x) = cos x − sin x − 2x +

0

0

(14.205) Taking Laplace transforms of both sides of (14.205) gives L{u (x)} = L{cos x − sin x − 2x} + L{1 ∗ (u2 (x) + v2 (x))}, L{v (x)} = L{− cos x − sin x + 2 cos2 x − 2} + L{1 ∗ (u2 (x) − v 2 (x))}, (14.206) so that 1 1 s−1 2 + 2 L{u2 (x) + v 2 (x)}, U (s) = + − s s(s2 + 1) s3 s (14.207) 1 1 s+1 2(s2 + 2) 2 2 2 + L{u (x) − v (x)}. V (s) = − + − s s(s2 + 1) s2 (s2 + 4) s2 s2 Proceeding as in the previous two examples we ﬁnd 3 1 1 1 5 x + ··· , u0 (x) = 1 + x − x2 − x3 + x4 + 2 6 24 120 1 1 u1 (x) = x2 − x4 − x5 + · · · , 6 6 1 4 1 5 u2 (x) = x + x + · · · , 6 6 and 1 1 1 1 v0 (x) = 1 − x − x2 − x3 + x4 + x5 + · · · , 2 2 24 8 2 3 1 4 1 5 (14.208) v1 (x) = x − x − x , · · · , 3 6 6 1 1 v2 (x) = x4 + x5 + · · · . 6 30 Using (14.182) we obtain the series solutions 1 1 1 1 u(x) = 1 − x2 + x4 + · · · + x − x3 + x5 + · · · , 2! 4! 3! 5! (14.209) 1 4 1 3 1 5 1 2 v(x) = 1 − x + x + · · · − x − x + x + · · · . 2! 4! 3! 5! Consequently, the exact solutions are given by (u(x), v(x)) = (cos x + sin x, cos x − sin x). (14.210) Example 14.28 Solve the system of nonlinear Volterra integro-diﬀerential equation by using the combined Laplace transform-Adomian decomposition method

14.4 Systems of Nonlinear Volterra Integro-Diﬀerential Equations

u (x) =

7 x 1 e − e2x − e4x + 3 3

2 1 v (x) = ex + 3e2x + e4x + 3 3

x

ex−t u2 (t) + v 2 (t) dt,

0

0

463

x

ex−t u2 (t) − v2 (t) dt,

(14.211)

u(0) = 1, u (0) = 1, v(0) = 1, v (0) = 2. Taking Laplace transforms of both sides of (14.211) gives 7 x 1 4x 2x + L{ex−t ∗ (u2 (x) + v2 (x))}, e −e − e L{u (x)} = L 3 3 1 4x 2 x 2x + L{ex−t ∗ (u2 (x) − v 2 (x))}, L{v (x)} = L e + 3e + e 3 3 (14.212) so that 1 7 − s2 U (s) − su(0) − u (0) = 3(s − 1) s − 2 1 1 − + L{u2 (x) + v 2 (x)}, 3(s − 4) s − 1 (14.213) 2 3 s2 V (s) − sv(0) − v (0) = + 3(s − 1) s − 2 1 1 + L{u2 (x) − v 2 (x)}, + 3(s − 4) s − 1 or equivalently 1 7 1 1 1 U (s) = 2 + + 2 − 2 − 2 s s 3s (s − 1) s (s − 2) 3s (s − 4) 1 L{u2 (x) + v 2 (x)}, + 2 s (s − 1) (14.214) 2 3 1 1 2 + 2 + 2 V (s) = 2 + + 2 s s 3s (s − 1) s (s − 2) 3s (s − 4) 1 + 2 L{u2 (x) − v 2 (x)}. s (s − 1) Proceeding as before we obtain 1 1 7 9 u0 (x) = 1 + x + x2 − x3 − x4 − x5 + · · · , 2 6 24 40 1 1 7 u1 (x) = x3 + x4 + x5 + · · · , 3 3 30 and

4 3 23 v0 (x) = 1 + 2x + 2x2 + x3 + x4 + x5 + · · · , 3 4 60 1 7 v1 (x) = − x4 − x5 , · · · . 12 60 Using (14.182) gives the series solutions by

(14.215)

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14 Nonlinear Volterra Integro-Diﬀerential Equations

1 2 1 1 1 x + x3 + x4 + x5 + · · · , 2! 3! 4! 5! 1 1 1 1 v(x) = 1 + 2x + (2x)2 + (2x)3 + (2x)4 + (2x)5 + · · · . 2! 3! 4! 5! Consequently, the exact solutions are given by (u(x), v(x)) = (ex , e2x ). u(x) = 1 + x +

(14.216)

(14.217)

Exercises 14.4.2 Use the combined Laplace transform-Adomian decomposition method to solve the following systems of nonlinear Volterra integro-diﬀerential equations by x ⎧ 1 6 ⎪ ⎪ u (x) = 2x − x2 − + (x − t)(u2 (t) + v 2 (t))dt x ⎪ ⎪ ⎪ 15 0 ⎨ x 1 4 1. x v (x) = + (x − t)(u2 (t) − v 2 (t))dt ⎪ ⎪ ⎪ 3 0 ⎪ ⎪ ⎩ u(0) = 1, v(0) = 1 x ⎧ 2 3 1 9 ⎪ 2 ⎪ (x) = 3x − − + (x − t)2 (u2 (t) + v 2 (t))dt x x u ⎪ ⎪ ⎪ 3 126 0 ⎨ x 1 7 2. 2 (x) = −3x − + (x − t)2 (u2 (t) − v 2 (t))dt v x ⎪ ⎪ ⎪ 35 0 ⎪ ⎪ ⎩ u(0) = 1, v(0) = 1 ⎧ x ⎪ ⎪ u (x) = −2 − 3x − 2 sin x + 3 cos x + (u2 (t) + v 2 (t))dt ⎪ ⎪ ⎪ 0 ⎨ x 1 3. (u2 (t) − v 2 (t))dt ⎪ v (x) = −2 + sin x + 2 cos x + sin(2x) + ⎪ ⎪ 2 0 ⎪ ⎪ ⎩ u(0) = 1, v(0) = 2 ⎧ x ⎪ ⎪ u (x) = cos x − 3 sin x + cos(x − t)(u2 (t) + v 2 (t))dt ⎪ ⎪ ⎪ 0 ⎨ x 1 4. sin(2x) − x + (x) = sin x + cos x + (x − t)(u2 (t) − v 2 (t))dt v ⎪ ⎪ ⎪ 2 0 ⎪ ⎪ ⎩ u(0) = 1, v(0) = −1 x ⎧ 8 x 1 1 ⎪ ⎪ ex−t (u2 (t) + v 2 (t))dt e + 2e2x − e4x − e6x + u (x) = ⎪ ⎪ 15 3 5 ⎪ ⎨ 0x 2 x 1 1 5. e + 3e3x − e4x + e6x + ex−t (u2 (t) − v 2 (t))dt ⎪ v (x) = ⎪ ⎪ 15 3 5 0 ⎪ ⎪ ⎩ u(0) = 1, v(0) = 1 x ⎧ ⎪ ⎪ (x) = − sin x − cos x + cos(x + t)(u2 (t) + v 2 (t))dt u ⎪ ⎪ ⎪ 0 ⎨ x 1 6. (u2 (t) − v 2 (t))dt ⎪ v (x) = − sin x − sin(2x) + ⎪ ⎪ 2 0 ⎪ ⎪ ⎩ u(0) = 1, u (0) = 0, v(0) = 0, v (0) = 1

References x ⎧ 3 x 1 −x ⎪ 2x ⎪ (x) = + − e + ex−t (u2 (t) + v(t))dt u e e ⎪ ⎪ ⎪ 2 2 0 ⎨ x 1 1 −2x 7. x −x (x) = − + e + + ex−t (u(t) + v 2 (t))dt (1 + 3x)e e v ⎪ ⎪ ⎪ 3 3 0 ⎪ ⎪ ⎩ u(0) = 1, u (0) = 1, v(0) = 1, v (0) = −1 x ⎧ 2 5 ⎪ 3 ⎪ x u (x) = −2x − 2x − + (u2 (t) + v 2 (t))dt ⎪ ⎪ 5 ⎪ 0 ⎨ x 2 1 8. (x − t)(u2 (t) − v 2 (t))dt v (x) = − x3 − x5 + ⎪ ⎪ ⎪ 3 5 0 ⎪ ⎪ ⎩ u(0) = u (0) = 1, u (0) = 2, v(0) = −v (0) = 1, v (0) = 2

465

References 1. J. Abdelkhani, Higher order methods for solving Volterra integro-diﬀerential equations of the ﬁrst kind, Appl. Math. Comput., 57 (1993) 97–101. 2. P. Linz, A simple approximation method for solving Volterra integro-diﬀerential equations of the ﬁrst kind, J. Inst. Math. Appl., 14 (1974) 211–215. 3. P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, (1985). 4. C. Baker, The Numerical Treatment of Integral Equations, Oxford University Press, London, (1977). 5. H.T. Davis, Introduction to Nonlinear Diﬀerential and Integral Equations, Dover, Publications, New York, (1962). 6. A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 7. R.K. Miller, Nonlinear Volterra Integral Equations, W.A. Benjamin, Menlo Park, CA, (1967). 8. K. Maleknejad and Y. Mahmoudi, Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-diﬀerential equations, Appl. Math. Comput. 145 (2003) 641–653. 9. W.E. Olmstead and R.A. Handelsman, Asymptotic solution to a class of nonlinear Volterra integral equations, II, SIAM J. Appl. Math., 30 (1976) 180–189. 10. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997).

Chapter 15

Nonlinear Fredholm Integral Equations

15.1 Introduction It was stated in Chapter 4 that Fredholm integral equations arise in many scientiﬁc applications. It was also shown that Fredholm integral equations can be derived from boundary value problems. Erik Ivar Fredholm (1866–1927) is best remembered for his work on integral equations and spectral theory. Fredholm was a Swedish mathematician who established the theory of integral equations and his 1903 paper in Acta Mathematica played a major role in the establishment of operator theory. The linear Fredholm integral equations and the linear Fredholm integro-diﬀerential equations were presented in Chapters 4 and 6 respectively. It is our goal in this chapter to study the nonlinear Fredholm integral equations of the second kind and systems of nonlinear Fredholm integral equations of the second kind. The nonlinear Fredholm integral equations of the second kind are characterized by ﬁxed limits of integration of the form b K(x, t)F (u(t))dt. (15.1) u(x) = f (x) + λ a

where the unknown function u(x) occurs inside and outside the integral sign, λ is a parameter, and a and b are constants. For this type of equations, the kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x) such as u2 (x), sin(u(x)), and eu(x) . In this chapter, we will mostly use degenerate or separable kernels. A degenerate or a separable kernel is a function that can be expressed as the sum of product of two functions each depends only on one variable. Such a kernel can be expressed in the form n K(x, t) = gi (x) fi (t). (15.2) i=1

Several analytic and numerical methods have been used to handle the nonlinear Fredholm integral equations. In this text we will use the direct A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

468

15 Nonlinear Fredholm Integral Equations

computation method, the Adomian decomposition method (ADM) combined with the modiﬁed decomposition method (mADM), and the successive substitution method to handle these equations. Systems of nonlinear Fredholm integral equations will be examined as well. For each type of equations we will select the proper methods that facilitate the computational work. The emphasis in this text will be on the use of these methods rather than proving theoretical concepts of convergence and existence. The theorems of existence, uniqueness, and convergence are important and can be found in the literature. The concern will be on the determination of the solutions u(x) of the nonlinear Fredholm integral equations and systems of these equations.

15.2 Existence of the Solution for Nonlinear Fredholm Integral Equations In this section we will present an existence theorem for the solution of nonlinear Fredholm integral equations. The proof of this theorem can be found in [1–2] among other references. The criteria is similar to the criteria presented in Chapter 13 for nonlinear Volterra integral equations. We ﬁrst rewrite the nonlinear Fredholm integral equation of the second kind by b G(x, t, u(t)) dt. (15.3) u(x) = f (x) + λ a

The speciﬁc conditions under which a solution exists for the nonlinear Fredholm integral equation are: (i) The function f (x) is bounded, |f (x)| < R, in a x b. (ii) The function G(x, t, u(t)) is integrable and bounded where |G(x, t, u(t))| < K, in a x, t b. (iii) The function G(x, t, u(t)) satisﬁes the Lipschitz condition |G(x, t, z) − G(x, t, z )| < M |z − z |.

(15.4)

(15.5)

Using the successive approximations method, it is proved in [1] that the series obtained by this method converges uniformly for all values of λ for 1 , (15.6) λ< k(b − a) R where k is the larger of the two numbers K 1 + |λ|K(b−a) and M .

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

469

15.2.1 Bifurcation Points and Singular Points When the nonlinear Fredholm integral equation includes a parameter λ, it is then obvious that the solution of the equation depends on λ. It is possible that λ has a bifurcation point. The bifurcation point is a value of the parameter λ, say λ0 , such that when λ changes through λ0 , then the number of real solutions will be changed. To illustrates this phenomenon, we consider the nonlinear Fredholm integral equation 1 u2 (t) dt. (15.7) u(x) = 3 + λ 0

The solution of this equation is √ (1 − 2λ) ± 1 − 12λ u(x) = . (15.8) 2λ 1 1 The bifurcation point in this problem is λ0 = 12 . For λ 12 , the nonlinear 1 Fredholm equation has two real solutions, but has no real solutions for λ > 12 . 1 This change of λ through the bifurcation point λ0 = 12 caused a change in the number and structure of solutions. However, for λ = 0, we obtain u(x) = 3 by substituting this value of λ in the integral equation itself, whereas u(x) is undeﬁned by substituting λ = 0 into (15.8). Accordingly, the point λ = 0 is called a singular point. More examples will be presented in the next section to address the two phenomena. The following conclusions can be made for nonlinear Fredholm integral equations: (i) The solution of the nonlinear equation may not be unique, there may be more than one solution. (ii) Concerning the bifurcation point, there may be one ore more bifurcation points. This will be seen in the forthcoming examples.

15.3 Nonlinear Fredholm Integral Equations of the Second Kind We begin our study on nonlinear Fredholm integral equations of the second kind of the form b K(x, t)F (u(t))dt, (15.9) u(x) = f (x) + λ a

where the kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x). The unknown function u(x), that will be determined, occurs inside and outside the integral sign. In what follows we will employ four distinct methods, namely the direct computation method, the series solution method, the Adomian decomposition method, and

470

15 Nonlinear Fredholm Integral Equations

the successive approximations method to handle Eq. (15.9). Other methods can be found in this text and in the literature.

15.3.1 The Direct Computation Method In this section, the direct computational method will be applied to solve the nonlinear Fredholm integral equations. The method was used before in Chapters 4 and 6. It approaches nonlinear Fredholm integral equations in a direct manner and gives the solution in an exact form and not in a series form. It is important to point out that this method will be applied for equations where the kernels are degenerate or separable of the form n K(x, t) = gk (x)hk (t). (15.10) k=1

Examples of separable kernels are x − t, xt2 , x3 − t3 , xt4 + x4 t, etc. The direct computation method can be applied as follows: 1. We ﬁrst substitute (15.10) into the nonlinear Fredholm integral equation b K(x, t)F (u(t))dt. (15.11) u(x) = f (x) + λ a

2. This substitution gives u(x) = f (x) b h1 (t) F (u(t))dt + λg2 (x) +λg1 (x) a

b

h2 (t) F (u(t))dt + · · · a

b

+λgn (x)

hn (t) F (u(t))dt. a

(15.12) 3. Each integral at the right side of (15.12) depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (15.12) becomes u(x) = f (x) + λα1 g1 (x) + λα2 g2 (x) + · · · + λαn gn (x), where

(15.13)

b

hi (t) u(t)dt, 1 i n.

αi =

(15.14)

a

4. Substituting (15.13) into (15.14) gives a system of n algebraic equations that can be solved to determine the constants αi , 1 i n. Using the obtained numerical values of αi into (15.13), the solution u(x) of the nonlinear Fredholm integral equation (15.11) follows immediately. It is interesting to point out that we may get more than one value for one or more of αi , 1 i n. This is normal because the equation is nonlinear

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

471

and the solution u(x) is not necessarily unique for nonlinear problems. Linear Fredholm integral equations give unique solutions under the existence conditions presented above. In what follows we present some examples to illustrate the use of this method. Example 15.1 Use the direct computation method to solve the nonlinear Fredholm integral equation 1 u(x) = a + λ u2 (t)dt, a > 0. (15.15) 0

The integral at the right side of (15.15) is equivalent to a constant because it depends only on a function of the variable t with constant limits of integration. Consequently, we rewrite (15.15) as u(x) = a + λα, where

α=

1

u2 (t)dt.

(15.16) (15.17)

0

To determine α, we substitute (15.16) into (15.17) to obtain 1 (a + λα)2 dt, α=

(15.18)

0

where by integrating the right side we ﬁnd λ2 α2 − (1 − 2λa)α + a2 = 0. Solving the quadratic equation (15.19) for α gives √ (1 − 2aλ) ± 1 − 4aλ α= . 2λ2 Substituting (15.20) into (15.16) leads to the exact solutions: √ 1 ± 1 − 4aλ u(x) = . 2λ The following conclusions can be made here:

(15.19)

(15.20)

(15.21)

1. Using λ = 0 into (15.15) gives the exact solution u(x) = a. However, u(x) is undeﬁned by using λ = 0 into (15.21). The point λ = 0 is called the singular point of Equation (15.15). 1 , Equation (15.21) gives only one solution u(x) = 2a. The 2. For λ = 4a 1 point λ = 4a is called the bifurcation point of the equation. This shows that 1 for λ < 4a , then Equation (15.15) gives two real solutions, but has no real 1 solutions for λ > 4a . 1 3. For λ < 4a , Equation (15.21) gives two exact real solutions. Example 15.2 Use the direct computation method to solve the nonlinear Fredholm integral equation

472

15 Nonlinear Fredholm Integral Equations

1

u(x) = x + λ

xtu2 (t)dt.

(15.22)

0

The integral at the right side of (15.22) is equivalent to a constant because it depends only on a function of the variable t with constant limits of integration. Consequently, we rewrite (15.22) as u(x) = (1 + λα)x, where

1

α=

tu2 (t)dt.

(15.23) (15.24)

0

To determine α, we substitute (15.23) into (15.24) to obtain 1 (1 + λα)2 t3 dt, α=

(15.25)

0

where by integrating the right side and solving the resulting equation for α we obtain √ (4 − 2λ) ± 4 1 − λ α= . (15.26) 2λ2 Substituting (15.26) into (15.23) leads to the exact solutions √ 2±2 1−λ u(x) = x. (15.27) λ We next consider the following three cases: 1. Using λ = 0 into (15.22) gives the exact solution u(x) = x. However, u(x) is undeﬁned by using λ = 0 into (15.27). Hence, λ = 0 is called the singular point of Equation (15.22). 2. For λ = 1, Equation (15.27) gives only one solution u(x) = 2x. Therefore, the point λ = 1 is called the bifurcation point of the equation. 3. For λ < 1, Equation (15.27) gives two exact real solutions. This is normal for nonlinear equations. Example 15.3 Use the direct computation method to solve the nonlinear Fredholm integral equation √ 1 3 xu2 (t)dt. (15.28) u(x) = +λ 12 −1 Proceeding as before, we rewrite (15.28) as √ 3 + λαx, u(x) = 12 where 1 α= u2 (t)dt. −1

To determine α, we substitute (15.29) into (15.30) to obtain

(15.29)

(15.30)

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

1

α= −1

473

√ 2 3 + λαt dt, 12

(15.31)

where by integrating the right side and solving the resulting equation for α we obtain √ 6 ± 2 9 − λ2 α= . (15.32) 8λ2 Substituting (15.32) into (15.29) leads to the exact solutions √ √ 3 6 ± 2 9 − λ2 u(x) = + x. (15.33) 12 8λ We consider the following three cases: √

1. Using λ = 0 into (15.28) gives the exact solution u(x) = 123 . However, u(x) is undeﬁned by using λ = 0 into (15.33). Hence, λ = 0 is called the singular point of equation (15.28). √ 2. For λ = ±3, Equation (15.33) gives two solutions u(x) = 123 ± 14 x. Consequently, there are two bifurcation points, namely ±3 for this equation. This shows that for −3 < λ < 3, then equation (15.15) gives two real solutions, but has no real solutions for λ > 3 or λ < −3. 3. For −3 < λ < 3, Equation (15.33) gives two exact real solutions. Example 15.4 Use the direct computation method to solve the nonlinear Fredholm integral equation 9 1 1 2 2 xt u (t)dt. (15.34) u(x) = x + 5 3 −1 Proceeding as before, we rewrite (15.34) as 9 1 u(x) = + α x, 5 3 where 1

α=

t2 u2 (t)dt.

(15.35)

(15.36)

−1

To determine α, we substitute (15.35) into (15.36) to obtain 2 1 9 1 α= (15.37) + α t4 dt, 5 3 −1 where by integrating the right side and solving the resulting equation for α we obtain 18 81 α= , . (15.38) 5 10 Substituting (15.38) into (15.35) leads to the two exact solutions 9 (15.39) u(x) = 3x, x. 2

474

15 Nonlinear Fredholm Integral Equations

Example 15.5 Use the direct computation method to solve the nonlinear Fredholm integral equation 1 5 u(x) = x + x t2 u3 (t)dt. (15.40) 6 0 Proceeding as before, we rewrite (15.40) as 5 + α x, u(x) = 6 where 1

α=

(15.41)

t2 u3 (t)dt.

(15.42)

0

To determine α, we substitute (15.41) into (15.42) to obtain 3 1 5 + α t5 dt, α= (15.43) 6 0 where by integrating the right side and solving the resulting equation for α we obtain √ 21 1 4 . (15.44) α = ,− ± 6 3 2 Substituting (15.44) into (15.41) leads to the three exact solutions √ √ 1 1 u(x) = x, − (1 − 21)x, − (1 + 21)x. (15.45) 2 2 Example 15.6 Use the direct computation method to solve the nonlinear Fredholm integral equation 1 1 11 x + x2 + (xt2 + x2 t)u2 (t)dt. (15.46) u(x) = 35 5 −1 Proceeding as before, we rewrite (15.46) as 11 1 u(x) = +α x+ + β x2 , 35 5 where 1

α= −1

t2 u2 (t)dt,

1

β=

tu2 (t)dt.

(15.47)

(15.48)

−1

To determine α and β, we substitute (15.47) into (15.48) and integrate the right side to ﬁnd 2 2 1 2 11 2 1 4 11 +α + +β , β = +α + β . (15.49) α= 5 35 7 5 5 35 5 Solving (15.49) for α and β we obtain √ √ 24 4 35 1 35 83 6 131 (α, β) = , , ,− , ∓ ,− ± . (15.50) 35 5 70 5 140 7 5 20

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

Substituting (15.50) into (15.47) leads to the four exact solutions √ √ 5 35 35 2 2 3 2 u(x) = x + x , x − x , x± ∓ x . 2 4 7 20

475

(15.51)

Example 15.7 Use the direct computation method to solve the nonlinear Fredholm integral equation 1 2 83 50 2 u(x) = − − x + x + (1 + xt + x2 t2 )u2 (t) dt. (15.52) 15 30 21 0 We rewrite (15.52) as 83 50 2 −β x+ + γ x2 , (15.53) u(x) = − + α − 15 30 21 where 1 1 1 α= u2 (t)dt, β = tu2 (t)dt, γ = t2 u2 (t)dt. (15.54) 0

0

0

To determine α,β and γ, we substitute (15.53) into (15.54), integrate the right side of the resulting equations, and by solving the resulting system we obtain 17 23 13 , , . (15.55) (α, β, γ) = 15 30 21 Substituting (15.55) into (15.53) leads to the exact solutions u(x) = 1 − 2x + 3x2 .

(15.56)

Example 15.8 Use the direct computation method to solve the nonlinear Fredholm integral equation 1 17 2 131 691 3 − x− x −x + u(x) = (1 + x2 t + xt2 )u2 (t) dt. (15.57) 210 630 120 0 We rewrite (15.57) as 691 17 131 +α − −γ x− − β x2 − x3 , (15.58) u(x) = 210 630 120 where 1 1 1 u2 (t)dt, β = tu2 (t)dt, γ = t2 u2 (t)dt. (15.59) α= 0

0

0

To determine α,β and γ, we substitute (15.58) into (15.59), integrate the right side of the resulting equations, and by solving the resulting system we obtain 79 17 61 (α, β, γ) = , , . (15.60) 210 120 630 Substituting (15.60) into (15.58) leads to the exact solutions

476

15 Nonlinear Fredholm Integral Equations

u(x) = 1 − x − x3 .

(15.61)

Exercises 15.3.1 In Exercises 1–6, use the direct computation method to solve the following nonlinear Fredholm integral equations. Find the singular point and the bifurcation point for each equation: 1 1 tu2 (t)dt 2. u(x) = 2 + λ t2 u2 (t)dt 1. u(x) = 4 + λ 3. u(x) = 1 + λ

0

−1

5. u(x) = 3 − λ

0

1

1

0

t2 u2 (t)dt

tu2 (t)dt

√ 1 3 +λ 4. u(x) = xu2 (t)dt 4 −1 1 5 6. u(x) = x + λ t2 u2 (t)dt 4 0

In Exercises 7–12, use the direct computation method so solve the following nonlinear Fredholm integral equations: 1 1 51 2 17 x + x+ 7. u(x) = 1 − x2 u3 (t)dt 8. u(x) = 1 − x(u2 (t) + u3 (t))dt 35 3 0 −1 1 437 2 14 (xt + x2 t2 )(u(t) − u2 (t))dt x− x + 9. u(x) = 15 420 0 π 7 10. u(x) = sin x + cos(x + t)u2 (t)dt 3 0 1 11 7 163 2 11. u(x) = − + x− x + (1 + xt + x2 t2 )u2 (t)dt 15 15 105 −1 π 1 π (1 + sin(x + t))u2 (t)dt 12. u(x) = cos x − + 3 2 0

15.3.2 The Series Solution Method In this section, the series solution method will be applied to handle nonlinear Fredholm integral equations. Recall that the generic form of Taylor series at x = 0 can be written as ∞ u(x) = a n xn . (15.62) n=0

We will assume that the solution u(x) of the nonlinear Fredholm integral equation 1 u(x) = f (x) + λ K(x, t)F (u(t))dt, (15.63) 0

exists and is analytic, and therefore possesses a Taylor series of the form given in (15.62), where the coeﬃcients an will be determined recurrently. Substituting (15.62) into both sides of (15.63) gives

15.3 Nonlinear Fredholm Integral Equations of the Second Kind ∞

1

n

an x = T (f (x)) +

n=0

or for simplicity we use a0 +a1 x+a2 x2 +· · · = T (f (x))+

K(x, t)F

0

∞

477

a n tn

dt,

(15.64)

n=0

0

1

K(x, t)F (a0 +a1 t+a2 t2 +· · · )dt, (15.65)

where T (f (x)) is the Taylor series for f (x). The integral equation (15.63) will be converted to a traditional integral in (15.64) or (15.65), where instead of integrating the nonlinear term F (u(x)), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. Proceeding as in previous chapters, we ﬁrst integrate the right side of the integral in (15.64) or (15.65), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x in both sides of the resulting equation to obtain a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (15.62). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then the obtained series can be used for numerical purposes. In this case, the more terms we determine, the higher accuracy level we achieve. Recall that the series solution method works eﬀectively if the solution u(x) is a polynomial. However, if u(x) is not a polynomial, then approximations to the coeﬃcients aj , j 0 will be used. Example 15.9 Solve the nonlinear Fredholm integral equation by using the series solution method 1 2309 2 1 7159 x+ x + (xt2 − x2 t)u2 (t)dt. (15.66) u(x) = 1 + 7560 2160 36 0 Using the series form (15.62) into both sides of (15.66) gives 2309 2 7159 x+ x a0 + a1 x + a2 x2 + · · · = 1 + 7560 2160 1

2 1 + (xt2 − x2 t) a0 + a1 t + a2 t2 + · · · dt, 36 0 (15.67) where by integrating the right side, collecting the like powers of x, and equating the coeﬃcients of like powers of x in both sides yields a0 = 1, 1, a1 = 1, 3611.190273, (15.68) a2 = 1, −4848.332424, an = 0, for n 3. The exact solutions are given by

478

15 Nonlinear Fredholm Integral Equations

u(x) = 1 + x + x2 , 1 + 3611.190273x − 4848.332424x2.

(15.69)

Example 15.10 Solve the nonlinear Fredholm integral equation by using the series solution method 1 41 76 2 1 u(x) = 2 − x + x − x3 + (xt − x2 t2 )u2 (t)dt. (15.70) 45 945 48 −1 Using the series form (15.62) into both sides of (15.70) gives 76 2 41 x − x3 a0 + a1 x + a2 x2 + · · · = 2 − x + 45 945 1

2 1 + (xt − x2 t2 ) a0 + a1 t + a2 t2 + · · · dt, 48 −1 (15.71) where by integrating the right side, collecting the like powers of x, and equating the coeﬃcients of like powers of x in both sides yields a0 = 2, 2, a1 = −1, 0.2924553739, a2 = 0, −173.6222619, a3 = −1, −1, (15.72) an = 0, for n 4. The exact solutions are given by (15.73) u(x) = 2 − x − x2 , 2 + 0.2924553739x − 173.6222619x2 − x3 . Example 15.11 Solve the nonlinear Fredholm integral equation by using the series solution method 1 1 1 x 2 u(x) = e + (3 − e ) + (x − t)u2 (t)dt. (15.74) 16 4 0 Substituting the series (15.62) into both sides of (15.74) gives a0 + a1 x + a2 x2 + a3 x3 + · · ·

2 1 1 1 (x − t) a0 + a1 t + a2 t2 + · · · dt. = ex + (3 − e2 ) + 16 4 0 Proceeding as before we ﬁnd 1 a0 = 1, a1 = 1, a2 = , 2! 1 1 1 a3 = , a4 = , · · · , an = . 3! 4! n! This gives the solution in a series form 1 1 1 u(x) = 1 + x + x2 + x3 + x4 + · · · . 2! 3! 4! Consequently, the exact solution is given by u(x) = ex .

(15.75)

(15.76)

(15.77) (15.78)

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

479

Example 15.12 We next consider the nonlinear Fredholm integral equation π π π 1 u(x) = cos x − sin x − (x − t)u2 (t)dt. + (1 + π) + 60 120 60 0 Substituting the series (15.62) into both sides of (15.79) gives a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · π π + (1 + π) = cos x − sin x − 60 120 π

2 1 + (x − t) a0 + a1 t + a2 t2 + · · · dt. 60 0 Proceeding as before we ﬁnd 1 a0 = 1, a1 = −1, a2 = − , 2! 1 1 1 a 3 = , a 4 = , a5 = − , · · · , 3! 4! 5! and so on. The solution in a series form is given by 1 2 1 4 1 3 1 5 u(x) = 1 − x + x + · · · − x − x + x + · · · , 2! 4! 3! 5! that converges to the exact solution u(x) = cos x − sin x.

(15.79)

(15.80)

(15.81)

(15.82)

(15.83)

Exercises 15.3.2 Solve the following nonlinear Fredholm integral equations by using the series solution method 1 1 241 1091 (x − t)u2 (t)dt 1. u(x) = − x − x2 + 240 1080 36 0 1 7537 1 119 1091 2 2. u(x) = (x − t)2 u2 (t)dt − x− x + 7560 120 1080 36 0 1 23 1 223 2 3. u(x) = 1 − (xt2 − x2 t)u2 (t)dt x+ x + 945 216 36 0 1 23 1 4. u(x) = 1 − (xt2 − x2 t)u2 (t)dt x + x2 + 630 48 −1 1 892 2357 1 + x − x2 − x3 + 5. u(x) = (x − t)u3 (t)dt 2310 945 18 0 1 1 1 1 (1 − e2 )x + (1 + e2 ) + 6. u(x) = ex + (x − t)u2 (t)dt 16 32 8 0 1 1 1 xtu2 (t)dt (1 + e2 )x + 7. u(x) = ex − 32 8 0

480

15 Nonlinear Fredholm Integral Equations

8. u(x) = cos x −

1 π + 112 56

1 0

u2 (t)dt

15.3.3 The Adomian Decomposition Method The Adomian decomposition method has been outlined before in previous chapters and has been applied to a wide class of linear Fredholm integral equations and linear Fredholm integro-diﬀerential equations. The method usually decomposes the unknown function u(x) into an inﬁnite sum of components that will be determined recursively through iterations as discussed before. The Adomian decomposition method will be applied in this chapter to handle nonlinear Fredholm integral equations. Although the linear term u(x) is represented by an inﬁnite sum of components, the nonlinear terms such as u2 , u3 , u4 , sin u, eu , etc. that appear in the equation, should be expressed by a special representation, called the Adomian polynomials An , n 0. Adomian introduced a formal algorithm to establish a reliable representation for all forms of nonlinear terms. The Adomian polynomials were introduced in Chapter 13. In what follows we present a brief outline for using the Adomian decomposition method for solving the nonlinear Fredholm integral equation b u(x) = f (x) + λ K(x, t)F (u(t))dt, (15.84) a

where F (u(t)) is a nonlinear function of u(x). The nonlinear Fredholm integral equation (15.84) contains the linear term u(x) and the nonlinear function F (u(x)). The linear term u(x) of (15.84) can be represented normally by the decomposition series ∞ un (x), (15.85) u(x) = n=0

where the components un (x), n 0 can be easily computed in a recursive manner as discussed before. However, the nonlinear term F (u(x)) of (15.84) should be represented by the so-called Adomian polynomials An by using the algorithm n 1 dn i An = λ ui , n = 0, 1, 2, · · · . (15.86) F n! dλn i=0

λ=0

Substituting (15.85) and (15.86) into (15.84) gives ∞ b ∞ un (x) = f (x) + λ K(x, t) An (t) dt. n=0

a

(15.87)

n=0

To determine the components u0 (x), u1 (x), . . ., we use the following recurrence relation

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

481

u0 (x) = f (x), b K(x, t)A0 (t)dt, u1 (x) = λ a

b

u2 (x) = λ

K(x, t)A1 (t)dt,

(15.88)

a

.. .

b

K(x, t)An (t)dt, n 0.

un+1 (x) = λ a

Recall that in the modiﬁed decomposition method, a modiﬁed recurrence relation is usually used, where f (x) is decomposed into two components f1 (x) and f2 (x), such that f (x) = f1 (x)+f2 (x). In this case the modiﬁed recurrence relation becomes in the form u0 (x) = f1 (x), b K(x, t)A0 (t)dt, u1 (x) = f2 (x) + λ

a b

u2 (x) = λ

K(x, t)A1 (t)dt,

(15.89)

a

.. .

b

K(x, t)An (t)dt, n 0.

un+1 (x) = λ a

Having determined the components, the solution in a series form is readily obtained. The obtained series solution may converge to the exact solution if such a solution exists, otherwise the series can be used for numerical purposes. The following remarks can be observed: (i) The convergence of the decomposition method has been examined in the literature by many authors. (ii) The decomposition method always gives one solution, although the solution of the nonlinear Fredholm equation is not unique. The decomposition method does not address the existence and uniqueness concepts. (iii) The modiﬁed decomposition method and the noise terms phenomenon can be used to accelerate the convergence of the solution. Generally speaking, the Adomian decomposition method is reliable and eﬀective to handle diﬀerential and integral equations. This will be illustrated by using the following examples. Example 15.13 Use the Adomian decomposition method to solve the nonlinear Fredholm integral equation

482

15 Nonlinear Fredholm Integral Equations

1

u(x) = a + λ

u2 (t)dt, a > 0.

(15.90)

0

The Adomian polynomials for the nonlinear term u2 (x) are given by A0 (x) = u20 (x), A1 (x) = 2u0 (x)u1 (x), (15.91) A2 (x) = 2u0 (x)u2 (x) + u21 (x), and so on. Substituting the series (15.85) and the Adomian polynomials (15.91) into the left side and the right side of (15.90) respectively we ﬁnd 1 ∞ ∞ un (x) = a + λ An (t)dt. (15.92) 0 n=0

n=0

Using the Adomian decomposition method we set 1 Ak (t)dt, k 0. u0 (x) = a, uk+1 (x) = λ

(15.93)

0

This in turn gives u0 (x) = a, u1 (x) = λ u2 (x) = λ u3 (x) = λ u4 (x) = λ

1 0 1 0

u20 (t)dt = λa2 , (2u0 (t)u1 (t))dt = 2λ2 a3 , (15.94)

1

(2u0 (t)u2 (t) +

0 1 0

u21 (t))dt

3 4

= 5λ a ,

(2u0 (t)u3 (t) + 2u1 (t)u2 (t))dt = 14λ4 a5 ,

.. . The solution in a series form is given by u(x) = a + λa2 + 2λ2 a3 + 5λ3 a4 + 14λ4 a5 + · · · ,

(15.95)

that converges to the exact solution √ 1 1 − 1 − 4aλ ,0 < λ . (15.96) u(x) = 2λ 4a It is clear that only one solution was obtained by using the Adomian decomposition method. However, by using the direct computation method we obtained two solutions for this nonlinear problem as shown in Example 1 that was presented before. Example 15.14 Use the Adomian decomposition method to solve the nonlinear Fredholm integral equation 1 u(x) = 1 + λ (1 − u(t) + u2 (t))dt. (15.97) 0

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

This equation can be rewritten as

u(x) = 1 + λ + λ

1

(−u(t) + u2 (t))dt.

483

(15.98)

0

Substituting the series assumption for u(x) and the Adomian polynomials for u2 (x) into the left side and the right side of (15.98) we ﬁnd 1 ∞ ∞ ∞ un (x) = 1 + λ + λ un (t) + An (t) dt. (15.99) − 0

n=0

n=0

n=0

Using the Adomian decomposition method we set 1 (−uk (t) + Ak (t))dt, k 0. u0 (x) = 1 + λ, uk+1 (x) = λ

(15.100)

0

This in turn gives u0 (x) = 1 + λ, 1 (−u0 (t) + u20 (t))dt = λ2 + λ3 , u1 (x) = λ 0 1 u2 (x) = λ (−u1 (t) + 2u0 (t)u1 (t))dt = λ3 + 3λ4 + 2λ5 , 0 1 (−u2 (t) + 2u0 (t)u2 (t) + u21 (t))dt = λ4 + 6λ5 + 10λ6 + 5λ7 , u3 (x) = λ 0 1 (−u3 (t) + 2u0 (t)u3 (t) + 2u1 (t)u2 (t))dt = λ5 + 10λ6 + · · · , u4 (x) = λ .. .

0

(15.101) The solution in a series form is given by (15.102) u(x) = 1 + λ + λ2 + 2λ3 + 4λ4 + 9λ5 + · · · , that converges to the exact solution 1 + λ − (1 − 3λ)(1 + λ) u(x) = . (15.103) 2λ It is clear again that only one solution was obtained by using the Adomian decomposition method. However, by using the direct computation method we can easily derive two solutions given by 1 + λ ± (1 − 3λ)(1 + λ) . (15.104) u(x) = 2λ It is worth noting that two bifurcation points appear at λ = −1 and λ = 13 . Example 15.15 Use the modiﬁed Adomian decomposition method to solve the nonlinear Fredholm integral equation 1 1 − ex+3 + ext u3 (t)dt. (15.105) u(x) = ex + x+3 0

484

15 Nonlinear Fredholm Integral Equations

Substituting the series (15.85) and the Adomian polynomials (15.86) into the left side and the right side of (15.105) respectively we ﬁnd 1 ∞ ∞ 1 − ex+3 x xt + un (x) = e + e An (t)dt, (15.106) x+3 0 n=0 n=0 where An are the Adomian polynomials for u2 (x) as shown above. Using the modiﬁed decomposition method we set u0 (x) = ex , 1 1 − ex+3 + ext A0 (t)dt, u1 (x) = x+3 0 1 (15.107) 1 − ex+3 + ext u30 (t)dt = 0, = x+3 0 1 ext Ak (t)dt = 0, k 1. uk+1 (x) = 0

This in turn gives the exact solution u(x) = ex ,

(15.108)

that satisﬁes the integral equation. It is worth noting that we did not use the Adomian polynomials. This is due to the fact that the modiﬁed decomposition method accelerates the convergence of the solution. Example 15.16 Use the modiﬁed Adomian decomposition method to solve the nonlinear Fredholm integral equation 1 2x(u2 (t) − tan2 (t))dt. (15.109) u(x) = sec x − 2x + 0

Substituting the series (15.85) and the Adomian polynomials (15.86) into the left side and the right side of (15.109) respectively we ﬁnd 1 ∞ ∞ 2 un (x) = sec x − 2x + xt An (t) − tan (t) dt. (15.110) 0

n=0

n=0

Using the modiﬁed decomposition method we set u0 (x) = sec x, 1 xt(A0 (t) − tan2 (t))dt = 0. u1 (x) = −2x +

(15.111)

0

Consequently, the exact solution is given by u(x) = sec x.

(15.112)

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

485

Exercises 15.3.3 In Exercises 1–6, solve the following nonlinear Fredholm integral equations by using the Adomian decomposition method 1 1 1. u(x) = 4 + λ tu2 (t)dt 2. u(x) = 2 + λ t2 u2 (t)dt 0

3. u(x) = 1 + λ

−1

5. u(x) = 3 − λ

0

1

1

0

√ 1 3 4. u(x) = xu2 (t)dt +λ 4 −1 1 5 6. u(x) = x + λ t2 u2 (t)dt 4 0

t2 u2 (t)dt

tu2 (t)dt

In Exercises 7–12, solve the nonlinear Fredholm integral equations by using the modiﬁed Adomian decomposition method π √e−1 1 2t π 7. u(x) = tan x − + dt 8. u(x) = x − 1 + dt 2 2 1 + u2 (t) 0 1 + u (t) 0 π π 4 x(u2 (t) − 1)dt 9. u(x) = sec x + −1 x+ 4 0 1 1 10. u(x) = cosh x + − x + (x − t)(u2 − sinh2 t)dt 2 0 1 1 (x − t)u2 (t)dt 11. u(x) = ln x − 2x + + 4 0 1 (x − t)(1 − u(t) + u2 (t))dt 12. u(x) = ln x − 4x + 1 + 0

15.3.4 The Successive Approximations Method The successive approximations method or the Picard iteration method was introduced before in Chapter 3. The method provides a scheme that can be used for solving initial value problems or integral equations. This method solves any problem by ﬁnding successive approximations to the solution by starting with an initial guess as u0 (x), called the zeroth approximation. As will be seen, the zeroth approximation is any selective real-valued function that will be used in a recurrence relation to determine the other approximations. The most commonly used values for the zeroth approximations are 0, 1, or x. Of course, other real values can be selected as well. Given Fredholm integral equation of the second kind b u(x) = f (x) + λ K(x, t)F (u(t))dt, (15.113) a

where u(x) is the unknown function to be determined, K(x, t) is the kernel, F (u(t)) is a nonlinear function of u(t), and λ is a parameter. The successive approximations method introduces the recurrence relation

486

15 Nonlinear Fredholm Integral Equations

u0 (x) = any selective real valued function, b K(x, t)un (t)dt, n 0. un+1 (x) = f (x) + λ

(15.114)

a

The question of convergence of un (x) for linear equation is justiﬁed by Theorem 1 presented in Chapter 3. However, for nonlinear Fredholm integral equation, it was proved in [1] that 1 , (15.115) λ< k(b − 1) R and M as shown where k is the larger of the two numbers K 1 + |λ|K(b−a) above. At the limit, the solution is determined by using the limit u(x) = lim un+1 (x). n→∞

(15.116)

The successive approximations method, or the Picard iteration method will be illustrated by studying the following examples. Example 15.17 Use the successive approximations method to solve the nonlinear Fredholm integral equation π π2 1 u(x) = cos x − tu2 (t)dt. (15.117) + 48 12 0 For the zeroth approximation u0 (x), we can select (15.118) u0 (x) = 1. The method of successive approximations admits the use of the iteration formula π 1 π2 un+1 (x) = cos x − + tu2n(t)dt, n 0. (15.119) 48 12 0 Substituting (15.118) into (15.119) we obtain π π2 1 tu20 (t)dt = cos(x) + 0.2056167584, + u1 (x) = cos x − 48 12 0 π 1 π2 + u2 (x) = cos x − tu21 (t)dt = cos(x) − 0.05115268549, 48 12 0 π 1 π2 + u3 (x) = cos x − tu22 (t)dt = cos(x) + 0.01812692764, 48 12 0 (15.120) π 2 π 1 tu23 (t)dt = cos(x) − 0.005907183842, + u4 (x) = cos x − 48 12 0 π 1 π2 + u5 (x) = cos x − tu24 (t)dt = cos(x) + 0.001983411200, 48 12 0 .. . and so on. Consequently, the solution u(x) of (15.117)

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

u(x) = lim un+1 (x) = cos x. n→∞

487

(15.121)

Unlike the direct computation method where we may get more than a solution, the successive substitution method gives only one solution for the nonlinear Fredholm problem. However, using the direct computation method gives the two solutions 32 u(x) = cos x, cos x + 2 . (15.122) π Example 15.18 Use the successive approximations method to solve the nonlinear Fredholm integral equation 1 x 2 1 x (e + 1) + xtu2 (t)dt. (15.123) u(x) = e − 192 48 0 For the zeroth approximation u0 (x), we can select (15.124) u0 (x) = 1. The method of successive approximations admits the use of the iteration formula 1 x 2 1 un+1 (x) = ex − (e + 1) + xtu2n (t)dt, n 0. (15.125) 192 48 0 Substituting (15.124) into (15.125) we obtain 1 x 2 1 x (e + 1) + xtu20 (t)dt = ex − 0.03327633383x, u1 (x) = e − 192 48 0 1 x 2 1 xtu21 (t)dt = ex − 0.0009901404827x, (e + 1) + u2 (x) = ex − 192 48 0 1 x 2 1 xtu22 (t)dt = ex − 0.00002962822377x, u3 (x) = ex − (e + 1) + 192 48 0 1 x 2 1 x u4 (x) = e − xtu23 (t)dt = ex − 0.000000886721005x, (e + 1) + 192 48 0 .. . , (15.126) and so on. Consequently, the solution u(x) of (15.123) is given by u(x) = lim un+1 (x) = ex . n→∞

(15.127)

It is worth noting that the direct computation method gives an additional solution to this equation given by u(x) = ex + (208 − 8e)x. (15.128) Example 15.19 Use the successive approximations method to solve the nonlinear Fredholm integral equation

488

15 Nonlinear Fredholm Integral Equations

π 5π 2 1 π − + t(u(t) + u2 (t))dt. 12 144 36 0 For the zeroth approximation u0 (x), we can select u(x) = sin x + 1 −

(15.129)

(15.130) u0 (x) = 1. The method of successive approximations admits the use of the iteration formula π 5π 2 1 π − + t(un (t) + u2n (t))dt, n 0. (15.131) un+1 (x) = sin x + 1 − 12 144 36 0 Substituting (15.130) into (15.131), and proceeding as before we obtain the approximations u1 (x) = sin x + 0.6696616927, u2 (x) = sin x + 0.8214573046, u3 (x) = sin x + 0.8997853785,

(15.132)

u4 (x) = sin x + 0.9426743063, u5 (x) = sin x + 0.9668710030, and so on. Consequently, the solution u(x) of (15.129) is given by u(x) = lim un+1 (x) = 1 + sin x. n→∞

(15.133)

The direct computation method gives an additional solution to this equation given by 18 4 1− . (15.134) u(x) = sin x − 2 − π π Example 15.20 Use the successive approximations method to solve the nonlinear Fredholm integral equation 1 143 1 tu2 (t)dt. (15.135) u(x) = ln x + + 144 36 0 For the zeroth approximation u0 (x), we can select u0 (x) = 1. (15.136) The method of successive approximations admits the use of the iteration formula 1 1 143 + un+1 (x) = ln x + tu2n (t)dt, n 0. (15.137) 144 36 0 Substituting (15.136) into (15.137), and proceeding as before we obtain the approximations

15.3 Nonlinear Fredholm Integral Equations of the Second Kind

u1 (x) = u2 (x) = u3 (x) = u4 (x) = u5 (x) = .. .

ln x + 1.006944444, ln x + 1.000097120, ln x + 1.000001349, ln x + 1.000000019, ln x + 1.000000000,

489

(15.138)

and so on. Consequently, the solution u(x) of (15.135) u(x) = lim un+1 (x) = 1 + ln x. n→∞

(15.139)

It is worth noting that the direct computation method gives an additional solution to this equation given by u(x) = 72 + ln x.

(15.140)

Exercises 15.3.4 Use the successive approximations method to solve the following nonlinear Fredholm integral equations: π 1 π2 t(1 + u2 (t))dt + 1. u(x) = sin x − 64 48 0 π 5π 1 π 1+ + 2. u(x) = sin x + 1 − t(u + u2 (t))dt 16 12 48 0 π 5π 2 1 7 + 3. u(x) = cos x + − t(u + u2 (t))dt 6 144 36 0 1 1 1 t(u + u2 (t))dt (127 − e2 ) + 4. u(x) = ex + 144 36 0 1 1 1 5. u(x) = ex + t(1 + u2 (t))dt (131 − e2 ) + 144 36 0 1 1 1 6. u(x) = xex − xtu2 (t)dt (3 + e2 )x + 288 36 0 1 1 1 7. u(x) = ex + xt2 u2 (t)dt (1 − e2 )x + 384 96 0 1 1 1 8. u(x) = ex + u3 (t)dt (1 − e3 ) + 288 96 0 π 1 1 2 x+ 9. u(x) = cos x − xu3 (t)dt 144 96 0 1 1 1 1 x+ + 10. u(x) = ln x − (x − t)u2 (t)dt 18 144 36 0 1 1 10279 tu2 (t)dt x+ 11. u(x) = x ln x + 10368 36 0

490 12. u(x) = x + ln x −

15 Nonlinear Fredholm Integral Equations 1 5 + 648 36

1 0

tu2 (t)dt

15.4 Homogeneous Nonlinear Fredholm Integral Equations Substituting f (x) = 0 into the nonlinear Fredholm integral equation of the second kind b K(x, t)F (u(t))dt, (15.141) u(x) = f (x) + λ a

gives the homogeneous nonlinear Fredholm integral equation of the second kind given by b u(x) = λ K(x, t)F (u(t))dt. (15.142) a

In this section we will focus our study on the homogeneous nonlinear Fredholm integral equation (15.142) for the speciﬁc case where the kernel K(x, t) is separable. The main goal for studying the homogeneous nonlinear Fredholm equation is to ﬁnd nontrivial solution, because the trivial solution u(x) = 0 is a solution of this equation. Moreover, the Adomian decomposition method is not applicable here because it depends mainly on assigning a non-zero value for the zeroth component u0 (x), and f (x) = 0 in this kind of equations. The direct computation method will be appropriate to be employed here to handle this kind of equations.

15.4.1 The Direct Computation Method The direct computation method was used before in this chapter. This method replaces the homogeneous nonlinear Fredholm integral equations by a single algebraic equation or by a system of simultaneous algebraic equations depending on the number of terms of the separable kernel K(x, t). As stated before, the direct computation method handles Fredholm integral equations, homogeneous or nonhomogeneous, in a direct manner and gives the solution in an exact form and not in a series form. The direct computation method will be applied in this section for the degenerate or separable kernels of the form n gk (x)hk (t). (15.143) K(x, t) = k=1

15.4 Homogeneous Nonlinear Fredholm Integral Equations

491

The direct computation method can be applied as follows: 1. We ﬁrst substitute (15.143) into the homogeneous nonlinear Fredholm integral equation the form b u(x) = λ K(x, t) F (u(t))dt. (15.144) a

2. This substitution leads to b b u(x) = λg1 (x) h1 (t) F (u(t))dt + λg2 (x) h2 (t) F (u(t))dt + · · · a a b hn (t) F (u(t))dt. +λgn (x) a

(15.145) 3. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Consequently, Equation (15.145) becomes u(x) = λα1 g1 (x) + λα2 g2 (x) + · · · + λαn gn (x), where

(15.146)

b

hi (t) F (u(t))dt, 1 i n.

αi =

(15.147)

a

4. Substituting (15.146) into (15.147) gives a system of n simultaneous algebraic equations that can be solved to determine the constants αi , 1 i n. Using the obtained numerical values of αi into (15.146), the solution u(x) of the homogeneous nonlinear Fredholm integral equation (15.142) follows immediately. Example 15.21 Solve the homogeneous nonlinear Fredholm integral equation by using the direct computation method π2 cos x sin tu2 (t)dt. (15.148) u(x) = λ 0

This equation can be rewritten as u(x) = αλ cos x, where

π 2

α=

sin tu2 (t)dt.

(15.149) (15.150)

0

Substituting (15.149) into (15.150) gives π2 sin t cos2 tdt, α = α2 λ2

(15.151)

0

that gives 1 2 2 α λ . (15.152) 3 Recall that α = 0 gives the trivial solution. For α = 0, we ﬁnd that α=

492

15 Nonlinear Fredholm Integral Equations

3 . λ2 This in turn gives the eigenfunction u(x) by 3 u(x) = cos x. λ Notice that λ = 0 is a singular point. α=

(15.153)

(15.154)

Example 15.22 Solve the homogeneous nonlinear Fredholm integral equation by using the direct computation method 1 u(x) = λ ex−tu2 (t)dt. (15.155) 0

This equation can be rewritten as u(x) = αλex , where

1

(15.156)

e−t u2 (t)dt.

(15.157)

Substituting (15.156) into (15.157) gives 1 2 2 et dt, α=α λ

(15.158)

α= 0

0

that gives

α = α2 λ2 (e − 1).

(15.159)

Recall that α = 0 gives the trivial solution. For α = 0, we ﬁnd that 1 . (15.160) α= 2 λ (e − 1) This in turn gives the eigenfunction u(x) by 1 u(x) = (15.161) ex . λ(e − 1) Example 15.23 Solve the homogeneous nonlinear Fredholm integral equation by using the direct computation method π2 u(x) = λ sin(x − 2t)u2 (t)dt. (15.162) 0

Notice that the kernel sin(x − 2t) = sin x cos 2t − cos x sin 2t is separable. Equation (15.162) can be rewritten as u(x) = αλ sin x − βλ cos x, (15.163) where

α= 0

π 2

cos 2tu2 (t)dt,

π 2

β=

sin 2tu2 (t)dt.

(15.164)

0

Substituting (15.163) into (15.164) and proceeding as before we obtain

15.4 Homogeneous Nonlinear Fredholm Integral Equations

493

λ2 π 2 λ2 (α − β 2 ), β = − (αβπ − 2α2 − 2β 2 ). 8 4 Solving this system for α and β gives 32 8π , β=− 2 2 . α=− 2 2 λ (π − 16) λ (π − 16) This in turn gives the eigenfunction u(x) by 8 u(x) = − (π sin x − 4 cos x). λ(π 2 − 16) α=−

(15.165)

(15.166)

(15.167)

Example 15.24 Solve the homogeneous nonlinear Fredholm integral equation by using the direct computation method 1 u(x) = λ (xt + t2 )(u(t) + u2 (t))dt. (15.168) −1

Equation (15.168) can be rewritten as u(x) = αλx + βλ, where

1

t(u(t) + u2 (t))dt,

α=

β=

−1

1

(15.169)

t2 (u(t) + u2 (t))dt.

(15.170)

0

Proceeding as before we ﬁnd * 5 2λ − 3 2λ − 3 2λ − 3 , β=− α = 0, ,− . 2 3 4λ 2λ2 4λ2 This in turn gives the exact solutions √ 2λ − 3 (2λ − 3)( 15x − 3) , . u(x) = − 2λ 12λ

(15.171)

(15.172)

Exercises 15.4.1 Use the direct computation method to solve the homogeneous nonlinear Fredholm integral equations π π 2 2 sin x cos tu2 (t)dt 2. u(x) = λ sin x cos t(u(t) + u2 (t))dt 1. u(x) = λ 3. u(x) = λ 5. u(x) = λ 7. u(x) = λ 9. u(x) = λ

0

π 0 1 0 π 2

0 π 0

cos x cos t(t + u2 (t))dt

4. u(x) = λ

ex−2t u3 (t)dt

6. u(x) = λ

cos(x − 2t)u2 (t)dt

8. u(x) = λ

cos(x + 2t)(1 + u2 (t))dt

10. u(x) = λ

0

1

ex−t (u(t) + u2 (t))dt

0 1

ex−2t (3t2 + e−2t u2 (t))dt

0 π 2

0

sin(x + t)u2 (t)dt

1

−1

(xt + t2 )u2 (t)dt

494

15 Nonlinear Fredholm Integral Equations

11. u(x) = λ

1 −1

(xt + x2 t2 )u2 (t)dt

12. u(x) = λ

1

0

(xt − xt2 )u2 (t)dt

15.5 Nonlinear Fredholm Integral Equations of the First Kind The standard form of the nonlinear Fredholm integral equations of the ﬁrst kind is given by b f (x) = K(x, t)F (u(t))dt, (15.173) a

where the kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x). The linear Fredholm integral equation of the ﬁrst kind is presented in Chapter 4 where the homotopy perturbation method was used for handling this type of equations. To determine a solution for the nonlinear Fredholm integral equation of the ﬁrst kind (15.173), we ﬁrst convert it to a linear Fredholm integral equation of the ﬁrst kind of the form b f (x) = K(x, t)v(t)dt, x ∈ D (15.174) a

by using the transformation v(x) = F (u(x)). We assume that F (u(x)) is invertible, then we can set

(15.175)

(15.176) u(x) = F −1 (v(x)). The linear Fredholm integral equation of the ﬁrst kind has been investigated in Chapter 4. An important remark has been reported in [3] and other references concerning the data function f (x). The function f (x) must lie in the range of the kernel K(x, t) [3]. For example, if we set the kernel by (15.177) K(x, t) = ex sin t. Then if we substitute any integrable function F (u(x)) in (15.173), and we evaluate the integral, the resulting f (x) must clearly be a multiple of ex [3]. This means that if f (x) is not a multiple of the x component of the kernel, then a solution for (15.173) does not exist. This necessary condition on f (x) can be generalized. In other words, the data function f (x) must contain components which are matched by the corresponding x components of the kernel K(x, t) Nonlinear Fredholm integral equation of the ﬁrst kind is considered illposed problem because it does not satisfy the following three properties: 1. Existence of a solution. 2. Uniqueness of a solution.

15.5 Nonlinear Fredholm Integral Equations of the First Kind

495

3. Continuous dependence of the solution u(x) on the data f (x). This property means that small errors in the data f (x) should cause small errors [4] in the solution u(x). The three properties were postulated by Hadamard [5]. Any problem that satisﬁes the three aforementioned properties is called well-posed problem. For any ill-posed problem, a very small change on the data f (x) can give a large change in the solution u(x). This means that nonlinear Fredholm integral equation of the ﬁrst kind may lead to a lot of diﬃculties. Several methods have been used to handle the linear and the nonlinear Fredholm integral equations of the ﬁrst kind. The Legendre wavelets, the augmented Galerkin method, and the collocation method are examples of the methods used to handle this equation. The methods that we used so far in this text cannot handle this kind of equations independently if it is expressed in its standard form (15.174). However, in this text, we will ﬁrst apply the method of regularization that received a considerable amount of interest, especially in solving ﬁrst order integral equations. We will second apply the homotopy perturbation method [6] to handle speciﬁc cases of the Fredholm integral equations where the kernel K(x, t) is separable. In what follows we will present a brief summary of the method of regularization and the homotopy perturbation method that will be used to handle the Fredholm integral equations of the ﬁrst kind.

15.5.1 The Method of Regularization The method of regularization was established independently by Phillips [7] and Tikhonov [8]. The method of regularization consists of replacing ill-posed problem by well-posed problem. The method of regularization transforms the linear Fredholm integral equation of the ﬁrst kind b K(x, t)v(t)dt, x ∈ D, (15.178) f (x) = a

to the approximation Fredholm integral equation b μvμ (x) = f (x) − K(x, t)vμ (t)dt, x ∈ D,

(15.179)

a

where μ is a small positive parameter. It is clear that (15.179) is a Fredholm integral equation of the second kind that can be rewritten 1 1 b K(x, t)vμ (t) dt, x ∈ D. (15.180) vμ (x) = f (x) − μ μ a

496

15 Nonlinear Fredholm Integral Equations

Moreover, it was proved in [1,3,9] that the solution vμ of Equation (15.180) converges to the solution v(x) of (15.178) as μ → 0 according to the following Lemma [10]: Lemma 15.1 Suppose that the integral operator of (15.178) is continuous and coercive in the Hilbert space where f (x), u(x), and vμ (x) are deﬁned, then: 1. |vμ | is bounded independently of μ, and 2. |vμ (x) − v(x)| → 0 when μ → 0. The proof of this lemma can be found in [3, 9–10]. In summary, by combining the method of regulariztion with any of the methods used before for solving Fredholm integral equation of the second kind, we can solve Fredholm integral equation of the ﬁrst kind (15.178). The method of regulariztion transforms the ﬁrst kind equation to a second kind equation. The resulting integral equation (15.180) can be solved by any method that was presented before in this chapter. The exact solution v(x) of (15.178) can thus be obtained by v(x) = lim vμ (x). (15.181) μ→0

In what follows we will present four illustrative examples where we will use the method of regulariztion to transform the ﬁrst kind integral equation to a second kind integral equation. The resulting equation will be solved by any appropriate method that we used before. Example 15.25 Combine the method of regulariztion and the direct computation method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 12 2ex−4t u2 (t)dt. (15.182) ex = 0

We ﬁrst set

v(x) = u2 (x), u(x) = ±

to carry out (15.182) into

x

e =

1 2

v(x),

2ex−4t v(t)dt.

(15.183)

(15.184)

0

Using the method of regularization, Equation (15.184) can be transformed to 1 1 x 1 2 x−4t vμ (x) = e − 2e vμ (t) dt. (15.185) μ μ 0 To use the direct computation method, Equation (15.185) can be written as α x 1 − e , (15.186) vμ (x) = μ μ where

15.5 Nonlinear Fredholm Integral Equations of the First Kind

1 2

α= 0

2e−4t vμ (t) dt.

497

(15.187)

To determine α, we substitute (15.186) into (15.187) to ﬁnd 1 2 1 α 2e−3t dt. α= − μ μ 0 Integrating the right side and solve to ﬁnd that

(15.188)

1

α= This in turn gives 1 vμ (x) = μ

2(1 − e 2 ) 1

2 − (2 + μ)e 2

.

(15.189)

1

1−

2(1 − e 2 ) 1

2 − (2 + μ)e 2

ex .

(15.190)

The exact solution v(x) of (15.185) can be obtained by 1

v(x) = lim vμ (x) =

ex+ 2

. 2(e − 1) Using (15.183) gives the exact solution of (15.182) by , 1 ex+ 2 . u(x) = ± 1 2(e 2 − 1) Two more solutions to Equation (15.182) are given by μ→0

u(x) = ±e2x .

1 2

(15.191)

(15.192)

(15.193)

Example 15.26 Combine the method of regulariztion and the direct computation method to solve the nonlinear Fredholm integral equation of the ﬁrst kind π π sin(x − t)u2 (t)dt. (15.194) sin x = 2 0 We ﬁrst set v(x) = u2 (x), u(x) = ± v(x), (15.195) to carry out (15.194) into π π sin x = sin(x − t)v(t)dt. (15.196) 2 0 Using the method of regularization, Equation (15.196) can be transformed to π 1 π sin(x − t)vμ (t) dt. (15.197) sin x − vμ (x) = 2μ μ 0 The resulting Fredholm integral equation of the second kind will be solved by the direct computation method. Equation (15.197) can be written as α β π vμ (x) = − sin x + cos x, (15.198) 2μ μ μ where

498

15 Nonlinear Fredholm Integral Equations

π

α= 0

cos t vμ (t) dt,

β= 0

1 2

sin t vμ (t) dt.

(15.199)

To determine α and β, we substitute (15.198) into (15.199), integrate the resulting integral and solve to ﬁnd that π3 π2 μ α= , β = . (15.200) 2(π 2 + 4μ2 ) π 2 + 4μ2 Substituting this result into (15.198) gives the approximate solution 2πμ π2 sin x + 2 cos x. (15.201) vμ (x) = 2 2 π + 4μ π + 4μ2 The exact solution v(x) of (15.197) can be obtained by v(x) = lim vμ (x) = cos x. μ→0

Using (15.195) gives the exact solution of (15.194) by √ u(x) = ± cos x.

(15.202)

(15.203)

Example 15.27 Combine the method of regulariztion and the Adomian decomposition method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 1 64 xtu4 (t)dt. (15.204) x= 15 −1 We ﬁrst set v(x) = u4 (x), u(x) = ± 4 v(x), (15.205) to carry out (15.204) into

1 64 x= xtv(t)dt. (15.206) 15 −1 Using the method of regularization, Equation (15.206) can be transformed to 1 1 64 x− xtvμ (t) dt. (15.207) vμ (x) = 15μ μ −1 The resulting Fredholm integral equation of the second kind will be solved by the Adomian decomposition method, where we ﬁrst set ∞ vμ (x) = vμn (x), (15.208) n=0

and the recurrence relation 64 1 1 x, vμk+1 (x) = − vμ0 (x) = xtvμk (t) dt, k 0. 15μ μ −1 This in turn gives the components 64 128 x, x, vμ1 (x) = − vμ0 (x) = 15μ 45μ2 256 512 x, vμ3 (x) = − x, vμ2 (x) = 135μ3 405μ4

(15.209)

(15.210)

15.5 Nonlinear Fredholm Integral Equations of the First Kind

499

and so on. Substituting this result into (15.208) gives the approximate solution 64 vμ (x) = x. (15.211) 5(2 + 3μ) The exact solution v(x) of (15.207) can be obtained by 32 v(x) = lim vμ (x) = x. (15.212) μ→0 5 Using (15.205) gives the exact solution * 4 32 x. (15.213) u(x) = ± 5 It is interesting to point out that there are two more solutions to this equation given by u(x) = ±(1 + x). (15.214) Example 15.28 Combine the method of regulariztion and the successive approximations method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 1 1 xtu3 (t)dt. (15.215) x= 5 0 We ﬁrst set v(x) = u3 (x), u(x) = ± 3 v(x), (15.216) to carry out (15.215) into

1 1 xtv(t)dt. (15.217) x= 5 0 Using the method of regularization, Equation (15.215) can be transformed to 1 1 1 xt vμ (t) dt. (15.218) vμ (x) = x− 5μ μ 0 To use the successive approximations method, we ﬁrst select uμ0 (x) = 0. Consequently, we obtain the following approximations 1 x, vμ0 (x) = 0, vμ1 (x) = 5μ 1 1 x, x− vμ2 (x) = 5μ 15μ2 (15.219) 1 1 1 x + x, vμ3 (x) = x− 5μ 15μ2 45μ3 1 1 1 1 vμ4 (x) = x− x+ x− x, 5μ 15μ2 45μ3 135μ4 and so on. Based on this we obtain the approximate solution 3 x. (15.220) vμ (x) = 5(1 + 3μ) The exact solution v(x) of (15.218) can be obtained by

500

15 Nonlinear Fredholm Integral Equations

v(x) = lim uμ (x) = μ→0

3 x. 5

Using (15.216) the exact solution is given by * 3 3 u(x) = x. 5 Another solution to this equation is given by u(x) = x.

(15.221)

(15.222) (15.223)

Exercises 15.5.1 Combine the regularization method with any method to solve the nonlinear Fredholm integral equations of the ﬁrst kind 1 1 1 3 1. e2x = e2x−3t u3 (t)dt 2. e−x = et−x u3 (t)dt 3 0 0 1 1 1 10 2 x2 t2 u3 (t)dt 4. xtu2 (t)dt 3. − x = x= 27 32 0 0 1 1 32 2 29 xtu2 (t)dt 6. x2 t2 u2 (t)dt 5. x= x = 36 63 0 −1 1 1 466 2 8 7. x2 t2 u2 (t)dt 8. xtu2 (t)dt x = x= 315 35 −1 −1 π π π π 9. cos x = cos(x − t)u2 (t)dt 10. sin x = cos(x − t)u2 (t)dt 2 2 0 0 π 1 22 π 709 sin(x − t)u2 (t)dt 12. cos(x − t)u3 (t)dt 11. − cos x = x− = 2 3 720 0 0

15.5.2 The Homotopy Perturbation Method In what follows we present the homotopy perturbation method for handling the nonlinear Fredholm integral equations of the ﬁrst kind of the form b K(x, t)v(t)dt. (15.224) f (x) = a

We ﬁrst deﬁne the operator

b

L(u) = f (x) −

K(x, t)u(t)dt = 0.

(15.225)

a

We next construct a convex homotopy of the form H(u, p) = (1 − p)u(x) + pL(u)(x) = 0.

(15.226)

The embedding parameter p monotonically increases from 0 to 1. The homotopy perturbation method admits the use of the expansion

15.5 Nonlinear Fredholm Integral Equations of the First Kind

u=

∞

p n un ,

501

(15.227)

n=0

and consequently v(x) = lim

∞

p→1

pn un (x).

(15.228)

n=0

The series (15.228) converges to the exact solution if such a solution exists. Substituting (15.227) into (15.226), and proceeding as in Chapter 4 we obtain the recurrence relation u0 (x) = 0, u1 (x) = f (x), b (15.229) K(x, t)un (t)dt, n 1. un+1 (x) = un (x) − a

If the kernel is separable, i.e. K(x, t) = g(x)h(t), then the following condition b K(t, t)dt < 1, (15.230) 1 − a must be justiﬁed for convergence. We will concern ourselves only on the case where K(x, t) = g(x)h(t). The HPM will be used to solve the following nonlinear Fredholm integral equations of the ﬁrst kind. Example 15.29 Use the homotopy perturbation method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 1 ex = ex−2t u2 (t)dt. (15.231) 0

We ﬁrst set

v(x) = u2 (x), u(x) = ±

to carry out (15.231) into

ex =

1

v(x),

ex−2t v(t)dt.

(15.232)

(15.233)

0

Notice that

1 −

0

1

K(t, t)dt = 0.3678 < 1.

Using the recurrence relation (15.229) we ﬁnd v0 (x) = 0, v1 (x) = ex , 1 vn+1 (x) = vn (x) − ex−2t vn (t)dt, n 1. 0

This in turn gives

(15.234)

(15.235)

502

15 Nonlinear Fredholm Integral Equations

v1 (x) = ex , 13 v2 (x) = v1 (x) − ex−2t v1 (t)dt = ex−1 , v0 (x) = 0,

v3 (x) = v2 (x) − v4 (x) = v3 (x) − v5 (x) = v4 (x) −

0

1 3

0 1 3

0 1 3

0

ex−2t v2 (t)dt = ex−2 ,

(15.236)

ex−2t v3 (t)dt = ex−3 , ex−2t v4 (t)dt = ex−4 ,

and so on. Consequently, the approximate solution is given by v(x) = ex (1 + e−1 + e−2 + e−3 + · · · ),

(15.237)

that converges to v(x) =

ex+1 . e−1

(15.238)

Using (15.232) gives the exact solution *

ex+1 . (15.239) e−1 It is worthnoting that there are two more solutions to this equation given by u(x) = ±ex . (15.240) u(x) = ±

The reason that the solution is not unique is due to the fact that the problem is nonlinear and ill-posed as well. Example 15.30 Use the homotopy perturbation method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 12 1 −x e = et−x u2 (t)dt. (15.241) 2 0 We ﬁrst set (15.242) v(x) = u2 (x), u(x) = ± v(x), to carry out (15.241) into 1 −x e = 2 Notice that

1 2

et−x v(t)dt.

(15.243)

0

12 K(t, t)dt = 0.5 < 1. 1 − 0

Using the recurrence relation (15.229) we ﬁnd

(15.244)

15.5 Nonlinear Fredholm Integral Equations of the First Kind

1 v1 (x) = e−x , 2 1 et−x vn (t)dt, n 1. vn+1 (x) = vn (x) −

503

v0 (x) = 0,

(15.245)

0

This in turn gives v0 (x) = 0,

1 −x e , 2

v1 (x) =

v2 (x) = v1 (x) −

1 3

0

et−x v1 (t)dt =

1 3

1 −x e , 4

1 −x e , 8 0 and so on. Consequently, the approximate solution is given by 1 1 1 1 −x v(x) = e + + + + ··· , 2 4 4 16 that converges to v(x) = e−x . v3 (x) = v2 (x) −

(15.246)

ex−2t v2 (t)dt =

(15.247) (15.248)

Using (15.242) gives the exact solution 1

u(x) = ±e− 2 x .

(15.249)

Example 15.31 Use the homotopy perturbation method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 1 3 − x= xtu3 (t)dt. (15.250) 8 0 We ﬁrst set 1 (15.251) v(x) = u3 (x), u(x) = v 3 (x), to carry out (15.250) into 3 − x= 8 Notice that

1 −

1 0

1

xtv(t)dt.

2 K(t, t)dt = < 1. 3

Using the recurrence relation (15.229) we ﬁnd 3 v0 (x) = 0, v1 (x) = − x, 8 vn+1 (x) = vn (x) − This in turn gives

(15.252)

0

1

0

xtvn (t)dt, n 1.

(15.253)

(15.254)

504

15 Nonlinear Fredholm Integral Equations

3 v1 (x) = − x, 8 1 1 v2 (x) = v1 (x) − xt v1 (t)dt = − x, 4 0 (15.255) 1 1 v3 (x) = v2 (x) − xt v2 (t)dt = − x, 6 0 1 1 xt v3 (t)dt = − x, v4 (x) = v3 (x) − 9 0 and so on. Consequently, the approximate solution is given by 2 4 8 3 + ··· , (15.256) v(x) = − x 1 + + + 8 3 9 27 that converges to 9 v(x) = − x. (15.257) 8 Using (15.251) gives the exact solution * 3 9 (15.258) − x. u(x) = 8 It is interesting to point out that there is one more solution to this equation given by u(x) = ln x. (15.259) v0 (x) = 0,

The reason that the solution is not unique is due to the fact that the problem is nonlinear and ill-posed as well. Example 15.32 Use the homotopy perturbation method to solve the nonlinear Fredholm integral equation of the ﬁrst kind 1 127 2 x = x2 t2 u2 (t)dt. (15.260) 252 0 We ﬁrst set (15.261) v(x) = u2 (x), u(x) = v(x), to carry out (15.260) into 127 2 x = 252 Notice that

1 −

0

1

1

x2 t2 v(t)dt.

(15.262)

0

2 K(t, t)dt = < 1. 3

(15.263)

Using the recurrence relation (15.229) and proceeding as before we ﬁnd

15.6 Systems of Nonlinear Fredholm Integral Equations

505

127 2 x , v1 (x) = 252 1 127 2 x , xtv1 (t)dt = v2 (x) = v1 (x) − 315 0 1 (15.264) 508 2 x , v3 (x) = v2 (x) − xtv2 (t)dt = 1575 0 1 2032 2 x , v4 (x) = v3 (x) − xtv3 (t)dt = 7875 0 and so on. Consequently, the approximate solution is given by 4 16 64 127 2 x 1+ + + + ··· , (15.265) v(x) = 252 5 25 125 that converges to 635 2 v(x) = x . (15.266) 252 Using (15.261) gives the exact solution * 635 x. (15.267) u(x) = ± 252 It is interesting to point out that there are two more solutions to this equation given by (15.268) u(x) = ±(x2 + x3 ). v0 (x) = 0,

The reason that the solution is not unique is due to the fact that the problem is nonlinear and ill-posed as well. Exercises 15.5.2 Use the homotopy perturbation method to solve the nonlinear Fredholm integral equations of the ﬁrst kind 1 1 1 3 1. e2x = e2x−3t u3 (t) dt 2. e−x = et−x u3 (t) dt 3 0 0 1 1 1 10 2 x2 t2 u3 (t) dt 4. xt u2 (t) dt 3. − x = x= 27 32 0 0 1 1 32 2 29 xt u2 (t) dt 6. x2 t2 u2 (t) dt 5. x= x = 36 63 0 −1 1 1 466 2 8 7. x2 t2 u2 (t) dt 8. xt u2 (t) dt x = x= 315 35 −1 −1

15.6 Systems of Nonlinear Fredholm Integral Equations In this section, we will study systems of nonlinear Fredholm integral equations of the second kind given by

506

15 Nonlinear Fredholm Integral Equations

b

˜ 1 (x, t)F˜1 (v(t)) dt, K1 (x, t)F1 (u(t)) + K

b

˜ 2 (x, t)F˜2 (v(t)) dt. K2 (x, t)F2 (u(t)) + K

u(x) = f1 (x) + a

v(x) = f2 (x) +

(15.269)

a

The unknown functions u(x) and v(x), that will be determined, occur inside ˜ i (x, t), and the funcand outside the integral sign. The kernels Ki (x, t) and K tion fi (x) are given real-valued functions, for i = 1, 2. The functions Fi and F˜i , for i = 1, 2 are nonlinear functions of u(x) and v(x). Systems of linear Fredholm integral equations were presented in Chapter 11. In this chapter, the system of nonlinear Fredholm integral equations can be handled by four distinct methods, namely the direct computation method, the modiﬁed Adomian method, the successive approximations method, and the series solution method. Although the aforementioned methods work effectively for handling the systems of nonlinear Fredholm integral equations, but only the ﬁrst two methods will be used in this section.

15.6.1 The Direct Computation Method The direct computation method will be applied to solve the systems of nonlinear Fredholm integral equations of the second kind. The method approaches Fredholm integral equations in a direct manner and gives the solution in an exact form and not in a series form. In what follows we summarize the necessary steps needed to apply this method. The method will be applied for the degenerate or separable kernels of the form n n ˜ k (t), ˜ 1 (x, t) = K1 (x, t) = gk (x)hk (t), K g˜k (x)h K2 (x, t) =

k=1 n

k=1

˜ 2 (x, t) = K

rk (x)sk (t),

k=1

n

(15.270) r˜k (x)˜ sk (t).

k=1

The direct computation method can be applied as follows: 1. We ﬁrst substitute (15.270) into the system (15.269) to obtain b n u(x) = f1 (x) + gk (x) hk (t)F1 (u(t)) dt +

n

k=1

k=1

˜ k (t)F˜1 (v(t)) dt, h

a

v(x) = f2 (x) + +

b

g˜k (x)

k=1

n

a

n k=1

(15.271)

b

rk (x)

sk (t)F2 (u(t)) dt a

b

s˜k (t)F˜2 (v(t)) dt.

r˜k (x) a

15.6 Systems of Nonlinear Fredholm Integral Equations

507

2. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (15.271) becomes u(x) = f1 (x) + α1 g1 (x) + · · · + αn gn(x) + β1 g˜1 (x) + · · · + βn g˜n (x), v(x) = f2 (x) + γ1 r1 (x) + · · · + γn rn (x) + δ1 r˜1 (x) + · · · + δn r˜n (x), (15.272) where b

hi (t) F1 (u(t)) dt, 1 i n,

αi = a

b

βi =

˜ i (t) F%1 (v(t))dt, 1 i n, h

a

(15.273)

b

si (t) F2 (u(t))dt, 1 i n,

γi = a

δi =

b

s˜i (t) F%2 (v(t))dt, 1 i n.

a

3. Substituting (15.272) into (15.273) gives a system of n algebraic equations that can be solved to determine the constants αi , βi , γi , and δi . To facilitate the computational work, we can use the computer symbolic systems such as Maple and Mathematica. Using the obtained numerical values of these constants into (15.272), the solutions u(x) and v(x) of the system of nonlinear Fredholm integral equations (15.269) follow immediately. The analysis presented above can be explained by studying the following examples. Example 15.33 Solve the following system of nonlinear Fredholm integral equations by using the direct computation method π u(x) = sin x + (1 − 2π) cos x + cos x(u2 (t) + v 2 (t)) dt, 0 (15.274) π (u2 (t) − v 2 (t)) dt. v(x) = sin x − cos x + 0

This system can be rewritten as u(x) = sin x + (1 − 2π + α + β) cos x, v(x) = sin x − cos x + (α − β), where π π u2 (t) dt,

α= 0

v 2 (t) dt.

β=

(15.275)

(15.276)

0

To determine α, and β, we substitute (15.275) into (15.276), and solving the resulting system, we obtain α = π, β = π. (15.277)

508

15 Nonlinear Fredholm Integral Equations

Substituting this result into (15.275) leads to the exact solutions (u(x), v(x)) = (sin x + cos x, sin x − cos x).

(15.278)

Example 15.34 Solve the following system of nonlinear Fredholm integral equations by using the direct computation method π4 π (tan x u2 (t) + sec x v 2 (t)) dt, u(x) = sec x − tan x + 4 0 (15.279) π4 π 2 2 v(x) = 1 − sec x + (tan x u (t) − sec x v (t)) dt. 4 0 Following the analysis presented above this system can be rewritten as π + β sec x + (α − 1) tan x, u(x) = 4 (15.280) π v(x) = 1 − − β sec x + α tan x, 4 where π4 π4 2 α= u (t) dt, β = v2 (t) dt. (15.281) 0

0

To determine α and β, we substitute (15.280) into (15.281) and solve the resulting system, we ﬁnd π α = 1, β = 1 − . (15.282) 4 Substituting (15.282) into (15.280) leads to the exact solutions (u(x), v(x)) = (sec x, tan x).

(15.283)

Example 15.35 Solve the following system of nonlinear Fredholm integral equations by using the direct computation method 1 x(u2 (t) + v 2 (t)) dt, u(x) = 1 − 6x + ln x + + 0 (15.284) 1 v(x) = 1 + 4x2 − ln x +

0+

x2 (u2 (t) − v 2 (t)) dt.

Proceeding as before, this system can be rewritten as u(x) = 1 + (α + β − 6)x + ln x, v(x) = 1 + (4 + α − β)x2 − ln x, where 1

α= 0+

u2 (t) dt,

1

β=

v2 (t) dt.

(15.285)

(15.286)

0+

To determine α, and β, we substitute (15.285) into (15.286), and proceeding as before to obtain α = 1, β = 5. (15.287)

15.6 Systems of Nonlinear Fredholm Integral Equations

509

This in turn gives the exact solutions (u(x), v(x)) = (1 + ln x, 1 − ln x).

(15.288)

Example 15.36 Solve the following system of nonlinear Fredholm integral equations by using the direct computation method π4 2 35(v 2 (t) − w2 (t)) dt, u(x) = 27 + sec x + 0

v(x) = 115 − sec2 x + w(x) = −3 − sec4 x +

π 4

0

0

π 4

35(w2 (t) − u2 (t)) dt, (u2 (t) − v 2 (t)) dt.

Proceeding as before, this system can be rewritten as u(x) = 27 + sec2 x + 35(β − γ), v(x) = 115 − sec2 x + 35(γ − α), w(x) = −3 − sec4 x + α − β, where π4 π4 π4 α= u2 (t) dt, β = v 2 (t) dt, γ = w2 (t)dt. 0

(15.289)

0

(15.290)

(15.291)

0

To determine α, β, and γ, we substitute (15.290) into (15.291) and by solving the resulting system we ﬁnd 10 π 2 π 8 π . (15.292) α= + , β= − , γ= + 4 3 4 3 4 105 These results lead to the exact solutions (u(x), v(x), w(x)) = (1 + sec2 x, 1 − sec2 x, 1 − sec4 x).

(15.293)

Exercises 15.6.1 Use the direct computation method to solve the following systems of nonlinear Fredholm 1 ⎧ integral equations

4 ⎪ ⎪ u(x) = x − + t u2 (t) + v 2 (t) dt ⎪ ⎨ 7 −1 1. 1 ⎪

4 ⎪ ⎪ ⎩ v(x) = x2 + x3 + + t u2 (t) − v 2 (t) dt 7 −1 ⎧ 1

⎪ ⎪ ln t u2 (t) + v 2 (t) dt ⎪ u(x) = 22 + ln x + ⎨ 2.

0+

⎪ ⎪ ⎪ ⎩ v(x) = −14 − ln x +

1 0+

ln t u2 (t) − v 2 (t) dt

510 15 Nonlinear Fredholm Integral Equations ⎧ 1 2

⎪ 2 ⎪ ⎪ ⎨ u(x) = 22 − 11 ln x + + ln(xt) u (t) + v (t) dt 0 3. 1 ⎪

⎪ ⎪ ⎩ v(x) = −14 + 7 ln x + ln(xt) u2 (t) − v 2 (t) dt 0+

⎧ π

π2 2 ⎪ ⎪ x + u(x) = sin x + cos x − xt u2 (t) + v 2 (t) dt ⎪ ⎨ 4 0 4. π ⎪

π ⎪ 2 ⎪ ⎩ v(x) = sin x − cos x − x + xt u2 (t) − v 2 (t) dt 2 0 ⎧ π 3

2 3π 2π ⎪ ⎪ − + u (t) + v 2 (t) dt ⎪ u(x) = x + sin2 x − ⎨ 4 3 0 5. π 3 ⎪

2π ⎪ ⎪ ⎩ v(x) = x − cos2 x − π 2 x + (x − t) u2 (t) − v 2 (t) dt + 3 0 ⎧ π √

3 ⎪ ⎪ tan2 t u2 (t) + sec2 t v 2 (t) dt u(x) = sec x − 2 3 + ⎪ ⎨ 0

6.

π ⎪ √

π ⎪ 3 ⎪ ⎩ v(x) = tan x − 2 3 + + sec2 t u2 (t) − tan2 t v 2 (t) dt 3 0 1 ⎧

11 ⎪ ⎪ u(x) = x+ (x − t) v 2 (t) + w2 (t) dt ⎪ ⎪ 35 ⎪ −1 ⎪ ⎪ ⎪ 1 ⎨

20 (x − 2t) w2 (t) + u2 (t) dt x+ v(x) = x2 − 7. ⎪ 12 −1 ⎪ ⎪ ⎪ 1 ⎪ ⎪

16 ⎪ ⎪ ⎩ w(x) = x3 − x+ (x − 3t) u2 (t) + v 2 (t) dt 15 −1 π ⎧

π 4 2 ⎪ ⎪ + v (t) − w 2 (t) dt u(x) = sec x tan x − 2 + ⎪ ⎪ 4 ⎪ 0 ⎪ ⎪ ⎪ π ⎨

π 4 8. w2 (t) − u2 (t) dt v(x) = sec2 x + 1 − + ⎪ 4 ⎪ 0 ⎪ ⎪ ⎪ π ⎪

⎪ 4 ⎪ ⎩ w(x) = sec2 x + u2 (t) − v 2 (t) dt 0

15.6.2 The modiﬁed Adomian Decomposition Method The modiﬁed Adomian decomposition method [11–12] was frequently and thoroughly used in this text. The method decomposes the linear terms u(x) and v(x) by an inﬁnite sum of components of the form ∞ ∞ un (x), v(x) = vn (x), (15.294) u(x) = n=0

n=0

where the components un (x) and vn (x) will be determined recurrently. The method can be used in its standard form, or combined with the noise terms phenomenon.

15.6 Systems of Nonlinear Fredholm Integral Equations

511

However, the nonlinear functions Fi and F˜i , for i = 1, 2, in (15.269) should be replaced by the Adomian polynomials An deﬁned by n 1 dn λi ui , n = 0, 1, 2, · · · , (15.295) An = F n! dλn i=0 λ=0

Substituting the aforementioned assumptions for the linear and the nonlinear terms into the system (15.269) and using the recurrence relations we can determine the components un (x) and vn (x). Having determined these components, the series solutions and the exact solutions are readily obtained. Example 15.37 Use the modiﬁed Adomian decomposition method to solve the following system of nonlinear Fredholm integral equations π

u(x) = sin x − π + (1 + xt)u2 (t) + (1 − xt)v 2 (t) dt, 0 (15.296) π

π2 2 2 x+ v(x) = cos x + (1 − xt)u (t) − (1 + xt)v (t) dt. 2 0 Substituting the linear terms u(x) and v(x) and the nonlinear terms u2 (x) and v 2 (x) from (15.294) and (15.295) respectively into (15.296) gives ∞ un (x) = sin x − π n=0 π ∞ ∞ + (1 + xt) An (t) + (1 − xt) Bn (t) dt, ∞ n=0

0

π2 x vn (x) = cos x + 2 π

+ 0

(1 − xt)

n=0

n=0

∞

∞

An (t) − (1 + xt)

n=0

Bn (t)

dt.

n=0

(15.297) The modiﬁed decomposition method will be used here, hence we set the recursive relation u0 (x) = sin x, v0 (x) = cos x, π

(1 + xt)u20 (t) + (1 − xt)v02 (t) dt = 0, u1 (x) = −π + (15.298) 0 π

π2 (1 − xt)u20 (t) − (1 + xt)v02 (t) dt = 0. x+ v1 (x) = 2 0 This in turn gives the exact solutions (u(x), v(x)) = (sin x, cos x).

(15.299)

Example 15.38 Use the modiﬁed Adomian decomposition method to solve the following system of Fredholm integral equations

512

15 Nonlinear Fredholm Integral Equations

u(x) = x + sec2 x −

8 π3 − + 3 96

2

v(x) = x − sec x − π + 2 ln 2 +

π 4

u2 (t) + v 2 (t) dt,

0

π 4

0

u (t) − v (t) dt. 2

(15.300)

2

Proceeding as before we obtain π ∞ ∞ ∞ 4 8 π3 2 + un (x) = x + sec x − − An (t) + Bn (t) dt, 3 96 0 n=0 n=0 n=0 π4 ∞ ∞ ∞ vn (x) = x − sec2 x − π + 2 ln 2 + An (t) = Bn (t) dt. 0

n=0

n=0

n=0

(15.301) For simplicity, the modiﬁed decomposition method will be used again, therefore we set the recursive relation u0 (x) = x + sec2 x, v0 (x) = x − sec2 x, π4

2 8 π3 + u1 (x) = − − u0 (t) + v02 (t) dt = 0, (15.302) 3 96 0 π4

2 u0 (t) + v02 (t) dt = 0. v1 (x) = −π + 2 ln 2 + 0

Consequently, the other components (uj , vj ) = (0, 0), j 2. As a result, the exact solutions are given by (15.303) (u(x), v(x)) = (x + sec2 x, x − sec2 x). Example 15.39 Use the modiﬁed Adomian decomposition method to solve the following system of nonlinear Fredholm integral equations 1 xt 2

2 x sinh(x + 2) + e u (t) + e−xt v2 (t) dt, u(x) = e − x+2 0 (15.304) 1 −xt 2

2 xt 2 −x e u (t) + e v (t) dt. sinh(x − 2) + v(x) = e − x−2 0 Proceeding as before and using the modiﬁed decomposition method, we ﬁnd the recursive relation u0 (x) = ex , v0 (x) = e−x , 1 xt 2

2 e u0 (t) + e−xt v02 (t) dt = 0, (15.305) sinh(x + 2) + u1 (x) = − x+2 0 1 −xt 2

2 e u0 (t) + ext v02 (t) dt = 0. sinh(x − 2) + v1 (x) = − x−2 0 The exact solutions are given by (15.306) (u(x), v(x)) = (ex , e−x ).

15.6 Systems of Nonlinear Fredholm Integral Equations

513

Example 15.40 Use the modiﬁed Adomian decomposition method to solve the following system of nonlinear Fredholm integral equations 1 x−3t 2

x x+1 u(x) = e − e e + v (t) + ex−6t w2 (t) dt, v(x) = e2x − 2ex + w(x) = e

3x

− 2e +

0 1

ex−6t w2 (t) + ex−2t u2 (t) dt,

(15.307)

0 1

x

ex−2t u2 (t) + ex−4t v 2 (t) dt.

0

Using the modiﬁed decomposition method, we set the recurrence relation u0 (x) = ex , v0 (x) = e2x , w0 (x) = e3x ,

u1 (x) = −ex+1 + v1 (x) = −2ex +

ex−3t v02 (t) + ex−6t w02 (t) dt = 0,

ex−6t w02 (t) + ex−2t u20 (t) dt = 0,

0

w1 (x) = −2ex +

0 1

1

1

(15.308)

ex−2t u2 (t) + ex−4t v 2 (t) dt = 0.

0

Consequently, the exact solutions are given by (u(x), v(x), w(x)) = (ex , e2x , e3x ).

(15.309)

Exercises 15.6.2 Use the modiﬁed Adomian decomposition method to solve the following systems of nonlinear Fredholm integral equations 1 ⎧

4 ⎪ ⎪ u(x) = x − + t u2 (t) + v 2 (t) dt ⎪ ⎨ 7 −1 1. 1 ⎪

4 ⎪ ⎪ t u2 (t) − v 2 (t) dt ⎩ v(x) = x2 + x3 + + 7 −1 ⎧ π

2 ⎪ ⎪ u (t) + v 2 (t) dt u(x) = sin x + cos x − 2π + ⎪ ⎨ 0 2. π ⎪

⎪ ⎪ xt u2 (t) − v 2 (t) dt ⎩ v(x) = sin x − cos x − πx + 0

⎧ ⎪ ⎪ ⎪ ⎨ u(x) = x + sin x + (4 − 2π) + 3.

π

0

2 u (t) − v 2 (t) dt

⎪ ⎪ ⎪ ⎩ v(x) = x − cos x + (8 − 4π − 2π 2 ) +

π

0

t u2 (t) − v 2 (t) dt

514

15 Nonlinear Fredholm Integral Equations

⎧ π

π−8 4 2 ⎪ ⎪ + u (t) + v 2 (t) dt ⎪ u(x) = sec x + ⎨ 4 0 4. π ⎪

π ⎪ 4 ⎪ ⎩ v(x) = tan x − x + x u2 (t) − v 2 (t) dt 4 0 ⎧ π

π 4 2 ⎪ ⎪ + u (t) − v 2 (t) dt u(x) = sec x − ⎪ ⎨ 4 0 5. π ⎪

π ⎪ 4 ⎪ ⎩ v(x) = tan x − x + x u2 (t) − v 2 (t) dt 4 0 √ ⎧ π

3 6 2 ⎪ ⎪ + u (t) − v 2 (t) dt u(x) = sec x tan x + ⎪ ⎨ 3 0 6. √ π ⎪

⎪ 3 11 6 ⎪ ⎩ v(x) = sec2 x − x u2 (t) + v 2 (t) dt x+ 27 0 ⎧ π 2

π + 4 4 ⎪ ⎪ u (t) + u(t)v(t) dt ⎪ ⎨ u(x) = sec x − 4 + 0 7. π ⎪

π − 4 ⎪ 4 2 ⎪ ⎩ v(x) = cos x + + u (t) − u(t)v(t) dt 4 0 ⎧ π

π−5 4 2 ⎪ ⎪ + u (t) + u(t)v 2 (t) dt u(x) = tan x + ⎪ ⎨ 4 0 8. π ⎪

π−3 ⎪ 4 2 ⎪ ⎩ v(x) = cos x + + u (t) − u(t)v 2 (t) dt 4 0 π ⎧

π − 10 4 2 ⎪ ⎪ u(x) = sec x + + v (t) + w 2 (t) dt ⎪ ⎪ 8 ⎪ 0 ⎪ ⎪ ⎪ π ⎨

π + 10 4 9. + w 2 (t) + u2 (t) dt v(x) = tan x − ⎪ 8 ⎪ 0 ⎪ ⎪ ⎪ π ⎪

⎪ π−8 4 2 ⎪ ⎩ w(x) = cos x + + u (t) + v 2 (t) dt 4 0 π ⎧

3π − 10π 2 4 2 ⎪ 2x+ ⎪ + v (t) + w 2 (t) dt u(x) = 1 + π sec ⎪ ⎪ 6 ⎪ 0 ⎪ ⎪ ⎪ π ⎨ 2

21π + 10π 4 10. v(x) = 1 − π sec2 x − + w2 (t) + u2 (t) dt ⎪ 6 ⎪ 0 ⎪ ⎪ ⎪ π ⎪ 2

⎪ π 3π + 16π 4 2 ⎪ ⎩ w(x) = 1 + sec2 x − + u (t) + v 2 (t) dt 2 6 0 π ⎧ 1 3 ⎪ ⎪ u(x) = sec x − + (v(t)w(t)) dt ⎪ ⎪ 2 ⎪ 0 ⎪ ⎪ ⎪ π ⎨ π 3 11. (w(t)u(t)) dt v(x) = tan x + + ⎪ 3 ⎪ 0 ⎪ ⎪ ⎪ π ⎪ ⎪ ⎪ ⎩ w(x) = cos x − 1 + 3 (u(t)v(t)) dt 0

References

515

π ⎧ 2−π 4 ⎪ ⎪ (v(t)w(t)) dt + u(x) = sec2 x + ⎪ ⎪ 8 ⎪ 0 ⎪ ⎪ ⎪ π ⎨ 1 4 12. (w(t)u(t)) dt v(x) = − sec2 x − + ⎪ 3 ⎪ 0 ⎪ ⎪ ⎪ π ⎪ ⎪ π ⎪ ⎩ w(x) = tan2 x − + 4 (u(t)v(t)) dt 4 0

References 1. H.T. Davis, Introduction to Nonlinear Diﬀerential and Integral Equations, Dover Publications, New York, (1962). 2. A. Jerri, Introduction to Integral Equations with Applications, Wiley, New York, (1999). 3. L.M. Delves and J. Walsh, Numerical Solution of Integral Equations,Oxford University Press, London, (1974). 4. R. Kress, Linear Integral Equations, Springer, Berlin, (1999). 5. J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Diﬀerential Equations, Yale University Press, New Haven, (1923). 6. J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg., 178 (1999) 257–262. 7. D.L. Phillips, A technique for the numerical solution of certain integral equations of the ﬁrst kind, J. Asso. Comput. Mach, 9 (1962) 84–96. 8. A.N. Tikhonov, On the solution of incorrectly posed problem and the method of regularization, Soviet Math, 4 (1963) 1035–1038. 9. R.F. Churchhouse, Handbook of Applicable Mathematics, Wiley, New York, (1981). 10. Y. Cherruault and V. Seng, The resolution of non-linear integral equations of the ﬁrst kind using the decomposition method of Adomian, Kybernetes, 26 (1997) 109–206. 11. A.M. Wazwaz, Partial Diﬀerential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009). 12. A.M. Wazwaz, A First Course in Integral Equations, World Scientiﬁc, Singapore, (1997).

Chapter 16

Nonlinear Fredholm Integro-Diﬀerential Equations

16.1 Introduction The linear Fredholm integral equations and the linear Fredholm integrodiﬀerential equations were presented in Chapters 4 and 6 respectively. In Chapter 15, the nonlinear Fredholm integral equations were examined. It is our goal in this chapter to study the nonlinear Fredholm integro-diﬀerential equations [1–7] and the systems of nonlinear Fredholm integro-diﬀerential equations. The nonlinear Fredholm integro-diﬀerential equations of the second kind is of the form 1 K(x, t)F (u(t)) dt, (16.1) u(n) (x) = f (x) + λ 0 (n)

where u (x) is the n th derivative of u(x). The kernel K(x, t) and the function f (x) are given real-valued functions, and F (u(x)) is a nonlinear function of u(x). In this chapter, we will mostly use degenerate or separable kernels. A degenerate or a separable kernel is a function that can be expressed as the sum of product of two functions each depends only on one variable. Such a kernel can be expressed in the form n gi (x) hi (t). (16.2) K(x, t) = i=1

Several analytic and numerical methods have been used to handle the nonlinear Fredholm integro-diﬀerential equations. In this text we will apply three of the methods used in this text, namely, the direct computation method, the variational iteration method (VIM), and the Taylor series solution method to handle the nonlinear Fredholm integro-diﬀerential equations. The emphasis in this text will be on the use of these methods rather than proving theoretical concepts of convergence and existence. The theorems of uniqueness, existence, and convergence are important and can be found in the literature. A-M. Wazwaz, Linear and Nonlinear Integral Equations © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

518

16 Nonlinear Fredholm Integro-Diﬀerential Equations

The concern will be on the determination of the solutions u(x) of nonlinear Fredholm integro-diﬀerential equations of the second kind.

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations The linear Fredholm integro-diﬀerential equation, where both diﬀerential and integral operators appear together in the same equation, has been studied in Chapter 6. In this section, the nonlinear Fredholm integro-diﬀerential equation will be examined. The standard form of the nonlinear Fredholm integrodiﬀerential equations of reads 1 K(x, t)F (u(t))dt, (16.3) u(i) (x) = f (x) + λ 0 (i)

di u dxi ,

and F (u(x)) is a nonlinear function of u(x) such as where u (x) = u2 (x), sin(u(x)), and eu(x) . Because the equation in (16.3) combines the differential operator and the integral operator, then it is necessary to deﬁne initial conditions u(0), u (0), . . . , u(i−1) (0) for the determination of the particular solution u(x) of this equation. In Chapter 6, we applied four methods to handle the linear Fredholm integro-diﬀerential equations of the second kind. In this section we will use only three of the methods that we used in Chapter 6. However, the other methods presented in Chapter 6 can be used as well. In what follows we will apply the direct computation method, the variational iteration method (VIM), and the series solution method to handle nonlinear Fredholm integro-diﬀerential equations of the second kind (16.3).

16.2.1 The Direct Computation Method The direct computation method has been extensively introduced in this text. Without loss of generality, we may assume a standard form to the Fredholm integro-diﬀerential equation given by b (n) u (x) = f (x) + K(x, t) F (u(t))dt, u(k) (0) = bk , 0 k (n − 1), (16.4) a

where u(n) (x) indicates the n th derivative of u(x) with respect to x, F (u(t)) is a nonlinear function of u(x), and bk are the initial conditions. It is important to point out that this method will be applied for equations where the kernels are degenerate or separable of the form n K(x, t) = gk (x)hk (t). (16.5) k=1

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations

519

Substituting (16.5) into the nonlinear Fredholm integro-diﬀerential equation (16.4) leads to b u(n) (x) = f (x) + g1 (x) h1 (t) F (u(t))dt a b b h2 (t) F (u(t))dt + · · · + gn (x) hn (t) F (u(t))dt. +g2 (x) a

a

(16.6) Each integral at the right side of (16.6) depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Based on this, Equation (16.6) becomes u(n) (x) = f (x) + α1 g1 (x) + α2 g2 (x) + · · · + αn gn (x), (16.7) where

b

hi (t) F (u(t))dt, 1 i n.

αi =

(16.8)

a

Integrating both sides of (16.7) n times, using the initial conditions, we obtain 1 1 xn−1 u(n−1) u(x) = u(0) + xu (0) + x2 u (0) + · · · + 2! (n − 1)! (16.9) +L−1 (f (x) + α1 g1 (x) + α2 g2 (x) + · · · + αn gn (x)) , −1

where L is the n-fold integral operator. Substituting (16.9) into (16.8) gives a system of n algebraic equations that can be solved to determine the constants αi , 1 i n. Using the obtained numerical values of αi into (16.7), the solution u(x) of the nonlinear Fredholm integro-diﬀerential equation (16.4) is readily obtained. It is interesting to point out that we may get more than one value for one or more of αi , 1 i n. This is normal because the equation is nonlinear and the solution u(x) may not be unique for nonlinear problems. In what follows we present some examples to illustrate the use of the direct computation method. Example 16.1 Solve the nonlinear Fredholm integro-diﬀerential equation by using the direct computation method π π2 xtu2 (t) dt, u(0) = 0. (16.10) u (x) = cos x − x+ 4 0 This equation may be written as π2 u (x) = cos x + α − x, u(0) = 0, (16.11) 4 where π tu2 (t) dt.

α=

(16.12)

0

Integrating both sides of (16.11) from 0 to x, and using the initial condition, we ﬁnd

520

16 Nonlinear Fredholm Integro-Diﬀerential Equations

u(x) = sin x +

α π2 − 2 8

x2 .

(16.13)

Substituting (16.13) into (16.12), evaluating the resulting integral, and solving the resulting equation for α we obtain π2 π 8 − 96π 3 + 576π + 96 , . (16.14) α= 4 4π 6 Consequently, the exact solutions are given by 12 72 12 u(x) = sin x, sin x − + 5 + 6 x2 . (16.15) π3 π π Example 16.2 Solve the nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 1 1 − e2 x+ xu2 (t) dt, u(0) = 1. (16.16) u (x) = ex + 2 0 This equation may be written as 1 − e2 + 2α x, u(0) = 1, (16.17) u (x) = ex + 2 where 1 α= u2 (t) dt. (16.18) 0

Integrating both sides of (16.17) from 0 to x, and by using the initial conditions we obtain 1 − e2 + 2α 2 x . u(x) = ex + (16.19) 4 Substituting (16.19) into (16.18) and proceeding as before we get e2 − 1 e2 − 40e + 119 , . (16.20) α= 2 2 The exact solutions are therefore given by u(x) = ex , ex + (30 − 10e)x2 .

(16.21)

Example 16.3 Solve the nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 19 11 1 1 u (x) = 2+ x+ x2 + (xt+x2 t2 ) (u(t)−u2 (t)) dt, u(0) = 1, u (0) = 1. 15 35 2 −1 (16.22) This equation may be written as 11 1 19 1 α+ x+ β+ x2 , u(0) = 1, u (0) = 1, (16.23) u (x) = 2 + 2 15 2 35 obtained by setting

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations

1

α= −1

t(u(t) − u2 (t)) dt,

1

β= −1

521

t2 (u(t) − u2 (t)) dt.

(16.24)

Integrating both sides of (16.23) two times from 0 to x, and by using the given initial conditions we ﬁnd 1 1 11 19 2 3 (16.25) α+ x + β+ x4 . u(x) = 1 + x + x + 12 90 24 420 Substituting (16.25) into (16.24), evaluating the integrals, and solving the resulting equations we ﬁnd 38 22 (16.26) α=− , β=− . 15 35 The exact solution is given by u(x) = 1 + x + x2 .

(16.27)

Example 16.4 Solve the nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 2 1 π cos(x − t) u2 (t) dt, u (x) = sin x + 3 2 0 (16.28) u(0) = 1, u (0) = 0, u (0) = −1. This equation may be written as 2 1 1 u (x) = sin x + α cos x + β sin x, 3 2 2 (16.29) u(0) = 1, u (0) = 0, u (0) = −1, obtained by setting π π 2 cos tu (t) dt, β = sin tu2 (t) dt. (16.30) α= 0

0

Integrating both sides of (16.29) three times from given initial conditions we obtain 2 1 1 1 u(x) = + β cos x + α(x − sin x) + β− 3 2 2 4 Proceeding as in the previous examples gives 2 α = 0, β = . 3 The exact solution is therefore given by u(x) = cos x.

0 to x, and by using the 1 6

1 1 x2 − β + . (16.31) 2 3 (16.32) (16.33)

Exercises 16.2.1 Solve the following nonlinear Fredholm integro-diﬀerential equations by using the direct computation method

522 1. u (x) = 1 −

16 Nonlinear Fredholm Integro-Diﬀerential Equations 1 17 x+ 24 2

1 0

xtu2 (t) dt, u(0) = 1

149 2 1 1 2 2 x tu (t) dt, u(0) = 1 x + 240 4 0 π 1 2 3. u (x) = cos x − x + xtu2 (t) dt, u(0) = 1 4 −π 2 1 2 1 π (π + 8)x + 4. u (x) = sin x + xt(u(t) − u2 (t)) dt, u(0) = 0 16 4 0 1 e e2 1 1 x(u(t) + u2 (t)) dt, u(0) = 1 + + x+ 5. u (x) = 1 + ex − 3 4 8 4 0 1 1 1 xt(1 + u(t) − u2 (t)) dt, u(0) = 2 6. u (x) = ex + (3 + e2 )x + 16 4 0 π2 1 π π x+ + 7. u (x) = − sin x − (x − t)u2 (t) dt, u(0) = 0, u (0) = 1 16 32 8 0 7 π 1 π sin(x − t)u2 (t) dt, u(0) = 0, u (0) = 0 8. u (x) = cos x + sin x + 3 2 2 0 1 1 1 x−2t 2 9. u (x) = ex + e u (t) dt, u(0) = 1, u (0) = 1 2 2 0 3 1 1 x−3t 3 10. u (x) = ex + e u (t) dt, u(0) = u (0) = u (0) = 1 4 4 0 1 1 x 1 11. u (x) = 8e2x − ex−4t u2 (t) dt, u(0) = 1, u (0) = 2, u (0) = 4 e + 16 16 0 π 1 π (u(t) − u2 (t)) dt, u(0) = −u (0) = 1, u (0) = 0 12. u (x) = sin x + + 16 8 0 2. u (x) = 1 + 2x −

16.2.2 The Variational Iteration Method The variational iteration method [8–10] was used before in previous chapters. The method handles linear and nonlinear problems in a straightforward manner. Unlike the Adomian decomposition method where we determine distinct components of the exact solution, the variational iteration method gives rapidly convergent successive approximations of the exact solution if such a closed form solution exists. The standard ith order nonlinear Fredholm integro-diﬀerential equation is of the form 1

u(i) (x) = f (x) +

K(x, t)F (u(t))dt,

(16.34)

0 i

where u(i) (x) = ddxui , and F (u(x)) is a nonlinear function of u(x). The initial conditions should be prescribed for the complete determination of the exact solution.

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations

523

The correction functional for the nonlinear integro-diﬀerential equation (16.34) is 1 t un+1 (x) = un (x) + λ(t) u(i) (t) − f (t) − K(t, r)F (˜ u (r)) dr dt. n n 0

0

(16.35) To apply this method in an eﬀective way, we should follow two essential steps: (i) It is required ﬁrst to determine the Lagrange multiplier λ that can be identiﬁed optimally via integration by parts and by using a restricted variation. The Lagrange multiplier λ may be a constant or a function. (ii) Having λ determined, an iteration formula, without restricted variation, should be used for the determination of the successive approximations un+1 (x), n 0 of the solution u(x). The zeroth approximation u0 can be any selective function. However, the initial values are preferably used for the selective zeroth approximation u0 . Consequently, the solution is given by u(x) = lim un(x). n→∞

(16.36)

The VIM will be illustrated by studying the following examples. Example 16.5 Use the variational iteration method to solve the nonlinear Fredholm integrodiﬀerential equation π 1 π u (x) = cos x − x + xu2 (t) dt, u(0) = 0. (16.37) 48 24 0 The correction functional for this equation is given by π x 1 π un (t) − cos t + t − tu2n (r)dr dt, (16.38) un+1 (x) = un (x) − 48 24 0 0 where we used λ = −1 for ﬁrst-order integro-diﬀerential equation. We can use the initial condition to select u0 (x) = u(0) = 0. Using this selection into the correction functional gives the following successive approximations u0 (x) = 0, u1 (x) = sin x − 0.03272492349x2, u2 (x) = sin x − 0.00663791983x2, (16.39) 2 u3 (x) = sin x − 0.00156723251x , u4 (x) = sin x − 0.00038016125x2, and so on. Using (16.36) gives the exact solution by u(x) = sin x.

(16.40)

Example 16.6 Use the variational iteration method to solve the nonlinear Fredholm integrodiﬀerential equation

524

16 Nonlinear Fredholm Integro-Diﬀerential Equations

π 1 π(π 2 + 3) x+ xtu2 (t) dt, u(0) = 0. (16.41) 512 64π 0 The correction functional for this equation is given by un+1 (x) = un(x) π x 1 π(π 2 + 3) un (t) − cos t + t sin t + t− − tru2n (r) dr dt. 512 64π 0 0 (16.42) We next select u0 (x) = 0 to ﬁnd the successive approximations u0 (x) = 0, u1 (x) = x cos x − 0.03948345181x2, u2 (x) = x cos x + 0.01017026496x2, (16.43) 2 u3 (x) = x cos x − 0.002418466659x , u4 (x) = x cos x + 0.000587237401x2, and so on. Proceeding as before, the exact solution is given by u(x) = x cos x. (16.44) u (x) = cos x − x sin x −

Example 16.7 Use the variational iteration method to solve the nonlinear Fredholm integrodiﬀerential equation π 1 3π x+ xu2 (t)dt, u(0) = 2, u (0) = 0. (16.45) u (x) = − cos x − 128 64 0 The correction functional for this equation is given by π x 1 3π t− (t − x) un (t) + cos t + tu2n (r) dr dt. un+1 (x) = un (x) + 128 64 0 0 (16.46) We can use the initial condition to select u0 (x) = u(0) + xu (0) = 2. Using this selection into the correction functional gives the following successive approximations u0 (x) = 2, u1 (x) = 1 + cos x + 0.02045307718x3, u2 (x) = 1 + cos x + 0.00118839931x3, (16.47) u3 (x) = 1 + cos x + 0.00004332607x3, u4 (x) = 1 + cos x + 0.00000152382x3, and so on. The VIM gives the exact solution by u(x) = 1 + cos x. (16.48) Example 16.8 Use the variational iteration method to solve the nonlinear Fredholm integrodiﬀerential equation

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations

u (x) = xex +

1079 x 1 e + 360 120

1

525

ex−2t u2 (t) dt, u(0) = 0, u (0) = 1, u (0) = 2.

0

(16.49) The correction functional for this equation is given by un+1 (x) = un (x) 1 1 1079 t 1 1 t t−2r 2 (t − x)2 u e (t) − te − − e u (r) dr dt. − n n 360 120 0 0 2 (16.50) We can use the initial condition to select u0 (x) = x + x2 . Using this selection into the correction functional gives the following successive approximations u0 (x) = x + x2 , u1 (x) = xex + (0.88212x + 0.441063x2 − 0.88212ex + 0.88212) × 10−3 , u2 (x) = xex + (0.267448x + 0.1337x2 − 0.26744ex + 0.26744) × 10−6 , u3 (x) = xex + (0.8115x + 0.40602x2 − 0.81286ex + 0.81135) × 10−10 , (16.51) and so on. The VIM admits the use of u(x) = lim un(x), (16.52) n→∞

that gives the exact solution by u(x) = xex .

(16.53)

Exercises 16.2.2 Use the variational iteration method to solve the nonlinear Fredholm integro-diﬀerential equations π π 1 1. u (x) = sin x − xu2 (t) dt, u(0) = 0 + 80 120 0 π 1 1 π 2. u (x) = cos x − x− + xu2 (t) dt, u(0) = 1 80 30 120 0 π π2 1 xtu2 (t) dt, u(0) = 0 (3 − π 2 )x + 3. u (x) = x + x cos x + 192 24 0 1 1 x 1 4. u (x) = 2e2x − ex−4t u2 (t) dt, u(0) = 1 e + 24 24 0 1 1 1 5. u (x) = −2e−2x − xex+4t u2 (t) dt, u(0) = 1 xex + 24 24 0 π 1 π 6. u (x) = cos x + sin x − + xu2 (t) dt, u(0) = −1 96 96 0 π π π2 1 (x − t)u2 (t) dt, u(0) = 0, u (0) = 1 x+ + 7. u (x) = − sin x − 36 72 18 0

526

16 Nonlinear Fredholm Integro-Diﬀerential Equations

8. u (x) = − cos x −

π2 1 x+ 288 72

π 0

xtu2 (t) dt, u(0) = 1, u (0) = 0

1 2 1 1 x(t − u2 (t)) dt, u(0) = 1, u (0) = 1 (e − 2)x + 4 2 0 π π 1 x(u(t) − u2 (t)) dt, u(0) = 1, u (0) = 0, u (0) = −1 10. u (x) = sin x + x+ 144 72 0 π 1 π 11. u (x) = sin x − cos x − x+ xu2 (t) dt, u(0) = 1, u (0) = 1, u (0) = −1 100 100 0 1 1 1 (e2 − 3)x + 12. u (x) = ex + x(2u(t) − u2 (t)) dt, u(0) = 2, u (0) = 200 100 0 1, u (0) = 1 9. u (x) = ex +

16.2.3 The Series Solution Method The series solution method depends mainly on the Taylor series for analytic functions [1]. A real function u(x) is called analytic if it has derivatives of all orders such that the generic form of Taylor series at x = 0 can be written as ∞ u(x) = a n xn . (16.54) n=0

The Taylor series method, or simply the series solution method will be used in this section for solving nonlinear Fredholm integro-diﬀerential equations of the second kind. We will assume that the solution u(x) of the nonlinear Fredholm integro-diﬀerential equation 1 (n) u (x) = f (x) + λ K(x, t)F (u(t))dt, u(k) (0) = k!ak , 0 k (n − 1), 0

(16.55) is analytic, and therefore possesses a Taylor series of the form given in (16.54), where the coeﬃcients an will be determined recurrently. The ﬁrst few coeﬃcients ak can be determined by using the initial conditions so that 1 1 (16.56) a0 = u(0), a1 = u (0), a2 = u (0), a3 = u (0), 2! 3! and so on. The remaining coeﬃcients ak of (16.54) will be determined by applying the series solution method to the nonlinear Fredholm integrodiﬀerential equation (16.55). Substituting (16.54) into both sides of (16.55) gives (n) ∞ ∞ 1 k k (16.57) ak x = T (f (x)) + K(x, t)F ak t dt, k=0

0

k=0

where T (f (x)) is the Taylor series for f (x). The integro-diﬀerential equation (16.55) will be converted to a traditional integral in (16.57) where instead of

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations

527

integrating the unknown function F (u(x)), terms of the form tn , n 0 will be integrated. Notice that because we are seeking series solution, then if f (x) includes elementary functions such as trigonometric functions, exponential functions, etc., then Taylor expansions for functions involved in f (x) should be used. We ﬁrst integrate the right side of the integral in (16.57), and collect the coeﬃcients of like powers of x. We next equate the coeﬃcients of like powers of x into both sides of the resulting equation to determine a recurrence relation in aj , j 0. Solving the recurrence relation will lead to a complete determination of the coeﬃcients aj , j 0, where some of these coeﬃcients will be used from the initial conditions. Having determined the coeﬃcients aj , j 0, the series solution follows immediately upon substituting the derived coeﬃcients into (16.54). The exact solution may be obtained if such an exact solution exists. If an exact solution is not obtainable, then a truncated series can be used for numerical purposes. In this case, the more terms we evaluate, the higher accuracy level we achieve. The following examples will be used to illustrate the series solution method. Example 16.9 Solve the nonlinear Fredholm integro-diﬀerential equation by using the series solution method 1 226 2 2 x + u (x) = 1 − x − (xt + x2 t2 )u2 (t))dt, u(0) = 1. (16.58) 15 105 −1 Substituting u(x) by the series ∞ an xn , (16.59) u(x) = n=0

into both sides of the equation (16.58) leads to ∞ 1 ∞ 226 2 2 n 2 2 n 2 x + an x = 1− x− an t ) dt. (16.60) (xt + x t )( 15 105 −1 n=0 n=0 Evaluating the integral at the right side, using a0 = 1, we ﬁnd a1 + 2a2 x + 3a3 x2 + 4a4 x3 + 5a5 x4 + · · · 4 4 4 4 4 2 4 = 1 + − + a1 + a3 a4 + a1 a2 + a1 a4 + a2 a3 + a3 x 15 3 9 5 7 7 5 4 2 2 4 4 2 2 2 2 52 2 2 4 − + a3 + a2 a4 + a1 a3 + a2 + a4 + a2 + a1 + a4 x2 . 35 9 9 7 7 7 5 5 11 (16.61) Equating the coeﬃcients of like powers of x in both sides, and solving the system of equations we obtain two sets of solutions a0 = 1, a1 = 1, a2 = 1, ar = 0, r 3, (16.62) Consequently, the exact solutions is given by u(x) = 1 + x + x2 ,

(16.63)

528

16 Nonlinear Fredholm Integro-Diﬀerential Equations

Example 16.10 Solve the nonlinear Fredholm integro-diﬀerential equation by using the series solution method 1 1 1 2 u (x) = ex − (e2 − 1)x + xu (t)dt, u(0) = 1. (16.64) 4 2 0 Substituting u(x) by the series ∞ u(x) = an xn , (16.65) n=0

into both sides of the equation (16.64) leads to ∞ 1 ∞ 1 2 n x n 2 an x = T1 (e − (e − 1)x) + (an t ) dt, x 4 0 n=0 n=0

(16.66)

where T1 is the Taylor series about x = 0. Evaluating the integral at the right side, using a0 = 1, and proceeding as before we ﬁnd 1 an = , n 0. (16.67) n! Using (16.65) gives the exact solution (16.68) u(x) = ex . It is worth noting that we used the series assumption up to O(x12 ) to get this result. Example 16.11 Solve the nonlinear Fredholm integro-diﬀerential equation by using the series solution method π 1 π π2 1 π + x+ + + u (x) = cos x − (x − t)u2 (t)dt, u(0) = 1. 8 3 6 16 12 0 (16.69) Substituting u(x) by the series ∞ an xn , (16.70) u(x) = n=0

into both sides of the equation (16.69) leads to ∞ π 1 π π2 n an x = T1 cos x − + x+ + 8 3 6 16 n=0 ⎛ ⎞ 2 π ∞ 1 ⎝(x − t) an tn ⎠ dt. + 12 0 n=0

(16.71)

Evaluating the integral at the right side, using a0 = 1, and proceeding as before we ﬁnd

16.2 Nonlinear Fredholm Integro-Diﬀerential Equations

529

a0 = 1, a2n+1 =

∞ n=0

(−1)n , n 0, (2n + 1)!

(16.72)

a2n = 0, n 0. Consequently, the series solution is given by ∞ (−1)n 2n+1 u(x) = 1 + x , (2n + 1)! n=0 that converges to the exact solution u(x) = 1 + sin x.

(16.73)

(16.74)

Example 16.12 Solve the nonlinear Fredholm integro-diﬀerential equation by using the series solution method 146 1 1 xtu2 (t)dt, u(0) = 1, u (0) = 0. (16.75) u (x) = 10 − x+ 35 2 −1 Substituting u(x) by the series ∞ u(x) = an xn , (16.76) n=0

into both sides of the equation (16.75), evaluating the integral at the right side, using a0 = a2 = 0, a1 = 1, and proceeding as before we ﬁnd a0 = 1, a1 = 0, a2 = 5, a3 = −1, ak = 0, k 4. Consequently, the exact solution is given by u(x) = 1 + 5x2 − x3 .

(16.77) (16.78)

Exercises 16.2.3 Solve the following nonlinear Fredholm integro-diﬀerential equations by using the series solution method 1 193 1 47 1. u (x) = − − x+ (x − t)u2 (t) dt, u(0) = 1 45 90 12 −1 1 53 1 197 (x − t)u2 (t) dt, u(0) = 1 − x + 3x2 + 2. u (x) = 45 630 12 −1 1 1 82 x+ 3. u (x) = 1 + xtu2 (t) dt, u(0) = 1 45 12 −1 1 1 2 1 xt(u2 (t) − u(t)) dt, u(0) = 1 (e − 3)x + 4. u (x) = ex − 96 24 0 1 1 x−1 1 5. u (x) = ex − + ex−2t (u2 (t) − u(t)) dt, u(0) = 1 e 24 24 0

530 6. u (x) = −2 −

16 Nonlinear Fredholm Integro-Diﬀerential Equations 1 4 x+ 15 2

1 −1

xtu2 (t) dt, u(0) = u (0) = 1

1 1 2 1 xtu2 (t) dt, u(0) = u (0) = 1 (e − 1)x + 48 12 0 1796 1 1 2 2 8. u (x) = xt u (t) dt, u(0) = 0, u (0) = 1 x+ 315 2 0

7. u (x) = ex −

16.3 Homogeneous Nonlinear Fredholm Integro-Diﬀerential Equations Substituting f (x) = 0 into the nonlinear Fredholm integral equation of the second kind b (i) K(x, t)F (u(t))dt, (16.79) u (x) = f (x) + λ a

gives the homogeneous nonlinear Fredholm integral equation of the second kind given by b u(i) (x) = λ K(x, t)F (u(t))dt, (16.80) a

where F (u(t)) is a nonlinear function of u(t). The initial conditions should be prescribed to determine the exact solution. In this section we will focus our study on the homogeneous nonlinear Fredholm integral equation (16.80) for the speciﬁc case where the kernel K(x, t) is separable. The aim for studying the homogeneous nonlinear Fredholm equation is to ﬁnd nontrivial solution. Moreover, the Adomian decomposition method is not applicable here because it depends mainly on assigning a nonzero value for the zeroth component u0 (x), and f (x) = 0 in this kind of equations. The direct computation method works eﬀectively to handle the homogeneous nonlinear Fredholm integro-diﬀerential equations.

16.3.1 The Direct Computation Method The direct computation method was used before in this chapter. This method replaces the homogeneous nonlinear Fredholm integro-diﬀerential equations by a single algebraic equation or by a system of simultaneous algebraic equations depending on the number of terms of the separable kernel K(x, t). The direct computation method handles Fredholm integro-diﬀerential equations, homogeneous or nonhomogeneous, in a direct manner and gives the solution in an exact form and not in a series form. As stated before, the direct computation method will be applied to equations where the kernel K(x, t0 ) is degenerate or separable kernel of the form

16.3 Homogeneous Nonlinear Fredholm Integro-Diﬀerential Equations

K(x, t) =

n

gk (x)hk (t).

531

(16.81)

k=1

The direct computation method can be applied as follows: 1. We ﬁrst substitute (16.81) into the homogeneous nonlinear Fredholm integro-diﬀerential equation b u(i) (x) = λ K(x, t) F (u(t))dt. (16.82) a

2. This substitution leads to b h1 (t) F (u(t))dt + λg2 (x) u(i) (x) = λg1 (x) a

b

h2 (t) F (u(t))dt + · · · a

b

+λgn (x)

hn (t) F (u(t))dt.

(16.83)

a

3. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that each integral is equivalent to a constant. Consequently, Equation (16.83) becomes u(i) (x) = λα1 g1 (x) + λα2 g2 (x) + · · · + λαn gn(x), where

b

hi (t) F (u(t))dt, 1 i n.

αi =

(16.84) (16.85)

a

4. Integrating both sides of (16.84) i times from 0 to x, and using the given initial conditions we obtain an expression for u(x) in terms of αi and x. 5. Substituting the resulting expression for u(x) into (16.85) gives a system of n simultaneous algebraic equations that can be solved to determine the constants αi , 1 i n. Using the obtained values of αi into (16.84), the solution u(x) of the integro-diﬀerential equation (16.80) follows immediately. Example 16.13 Solve the homogeneous nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 1 1 xu2 (t) dt, u(0) = 1. (16.86) u (x) = λ 12 0 This equation may be written as 1 u (x) = λαx, u(0) = 1, (16.87) 12 where 1 α= u2 (t) dt. (16.88) 0

Integrating both sides of (16.87) from 0 to x, and using the initial condition, we ﬁnd 1 (16.89) u(x) = 1 + λαx2 . 24

532

16 Nonlinear Fredholm Integro-Diﬀerential Equations

Substituting (16.89) into (16.88), evaluating the integral, and solving the resulting equation for α we obtain √ 2880 − 80λ ± 32 8100 − 450λ − 5λ2 α= . (16.90) 2λ2 This in turn gives the exact solutions √ 2880 − 80λ ± 32 8100 − 450λ − 5λ2 2 (16.91) x . u(x) = 1 + 48λ We now consider the following three cases: 1. Using λ = 0 into (16.89) gives the exact solution u(x) = 1. However, u(x) is undeﬁned by using λ = 0 into (16.91). Hence, λ = 0 is a singular point of homogeneous nonlinear √ Fredholm integro-diﬀerential equation (16.86). 2. For λ = −45 ± 27 5, Equation (16.91) gives the exact solutions √ u(x) = 1 ± 5 x2 . (16.92) √ Consequently, there are two bifurcation√points, namely −45 √ ± 27 5 for this equation. This shows that for −45 − 27 5 < λ < −45 + 27 5, then equation √ (16.86) gives two√real solutions, but has no real solutions for λ > −45 + 27 5 or λ < −45 − 27 5. √ √ 3. For −45 − 27 5 < λ < −45 + 27 5, Equation (16.86) gives two exact real solutions. This is normal for nonlinear problems, where solution may not be unique. Example 16.14 Solve the homogeneous nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 1 u (x) = λ ex−t u2 (t) dt, u(0) = 0. (16.93) 0

This equation can be rewritten as u (x) = αλex , u(0) = 0, where

1

α=

e−t u2 (t) dt.

(16.94) (16.95)

0

Integrating both sides of (16.94) from 0 to x, and using the initial condition, we ﬁnd u(x) = λα(ex − 1). (16.96) Substituting (16.96) into (16.95), evaluating the integral, and solving the resulting equation for α we obtain 1 . (16.97) α= 2 λ (2 sinh 1 − 2) This in turn gives the exact solution by ex − 1 . (16.98) u(x) = λ(2 sinh 1 − 2)

16.3 Homogeneous Nonlinear Fredholm Integro-Diﬀerential Equations

533

Notice that λ = 0 is a singular point of this equation. Example 16.15 Solve the homogeneous nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 1 u (x) = λ (xt + x2 t2 ) u2 (t) dt, u(0) = 0. (16.99) −1

This equation can be rewritten as u (x) = αλx + βλx2 , u(0) = 0, where

1

2

t u (t) dt,

α=

1

β=

−1

t2 u2 (t) dt.

(16.100) (16.101)

−1

Integrating both sides of (16.100) from 0 to x, and using the initial condition, we ﬁnd 1 1 u(x) = λαx2 + λβx3 . (16.102) 2 3 Substituting (16.102) into (16.101), evaluating the integrals, and solving the resulting system of equations for α and β we obtain √ 14 5 81 21 α = 0, , β = 2, 2. (16.103) 3λ2 2λ 2λ This in turn gives the exact solutions √ 27 3 7 5 2 7 3 x , x + x . u(x) = (16.104) 2λ 3λ 2λ Notice that λ = 0 is a singular point of this equation. Example 16.16 Solve the homogeneous nonlinear Fredholm integro-diﬀerential equation by using the direct computation method 1 (1 + xt) (u(t) − u2 (t)) dt, u(0) = u (0) = 0. (16.105) u (x) = λ −1

This equation can be rewritten as u (x) = αλ + βλx, u(0) = u (0) = 0, where 1 1 α= (u(t) − u2 (t)) dt, β = t (u(t) − u2 (t)) dt. −1

(16.106)

(16.107)

−1

Integrating both sides of (16.106) twice from 0 to x, and using the initial conditions, we ﬁnd 1 1 (16.108) u(x) = λαx2 + λβx3 . 2 6 Substituting (16.108) into (16.107), evaluating the integrals, and solving the resulting system of equations for α and β we obtain

534

16 Nonlinear Fredholm Integro-Diﬀerential Equations

7 15(29λ + 165)(λ − 15) 10(λ − 3) 7(λ − 15) , , β = 0, ± . (16.109) 3λ2 5λ2 25λ2 (i) For α = 10(λ−3) 3λ2 , β = 0, the exact solution is given by 5(λ − 3) 2 x . u(x) = (16.110) 3λ √

α=

7

15(29λ+165)(λ−15)

,β = ± , the exact solutions are (ii) For α = 7(λ−15) 5λ2 25λ2 given by 7 15(λ − 15)x2 ± 15(29λ + 165)(λ − 15) x3 . u(x) = (16.111) 150λ We now consider the following cases: 1. Using λ = 0 into (16.105) gives the trivial solution u(x) = 0. However, u(x) is undeﬁned by using λ = 0 into the last solution. Hence, λ = 0 is a singular point of equation (16.105). 2. There are two bifurcation points, namely λ = − 169 29 and λ = 15 for this equation. This shows that for λ < − 169 , and for λ > 15, then equation 29 (16.105) gives two real solutions, but has no real solutions for − 169 29 < λ < 15. Exercises 16.3.1

Use the direct computation method to solve the following Fredholm integro-diﬀerential equations: 1 1 x(u(t) − u3 (t)) dt, u(0) = 0 1. u (x) = λ 12 −1 1 1 λ 2. u (x) = x(u(t) − u2 (t)) dt, u(0) = 0 12 −1 1 1 x(1 − u2 (t)) dt, u(0) = 1 λ 3. u (x) = 24 0 π 1 4. u (x) = λ sin x sin t(1 + u2 (t)) dt, u(0) = 0 24 0 π 1 (x + x2 t)u2 (t) dt, u(0) = 0 5. u (x) = λ 12 0 π 6. u (x) = λ sin(x + t)(1 − u2 (t)) dt, u(0) = 0 7. u (x) = λ 8. u (x) = λ

9. u (x) = λ

−π π −π π −π

1

−1

cos(x + t)(1 − u2 (t)) dt, u(0) = 0 cos(x + t)(1 − u2 (t)) dt, u(0) = 0 x(u(t) + u2 (t)) dt, u(0) = 1, u (0) = 0

16.4 Systems of Nonlinear Fredholm Integro-Diﬀerential Equations 10. u (x) =

1 λ 2

1 λ 2 12. u (x) = λ

11. u (x) =

1

−1

1

−1 π

−π

535

x2 (1 + u2 (t)) dt, u(0) = 0, u (0) = 1 (1 − x2 t)(u(t) − u2 (t)) dt, u(0) = u (0) = 0

cos(x − t)(u(t) − u2 (t)) dt, u(0) = u (0) = 0

16.4 Systems of Nonlinear Fredholm Integro-Diﬀerential Equations In this section, systems of Fredholm integro-diﬀerential equations of the second kind given by b (i) ˜ 1 (x, t)F˜1 (v(t)) dt, u (x) = f1 (x) + K1 (x, t)F1 (u(t)) + K a (16.112) b ˜ 2 (x, t)F˜2 (v(t)) dt, K2 (x, t)F2 (u(t)) + K v(i) (x) = f2 (x) + a

will be studied. The unknown functions u(x) and v(x) occur inside the integral sign whereas the derivatives of u(x), v(x) appear mostly outside the ˜ i (x, t), and the function fi (x) are integral sign. The kernels Ki (x, t) and K given real-valued functions. The functions Fi and F˜i are nonlinear functions for u(x) and v(x) respectively. In Chapter 11, two analytical methods were used for solving systems of linear Fredholm integro-diﬀerential equations. These methods are the direct computation method and the variational iteration method. The aforementioned methods can eﬀectively handle the systems of nonlinear Fredholm integro-diﬀerential equations (16.112). The other methods presented in this text can also be used for handling such systems.

16.4.1 The Direct Computation Method The direct computation method will be applied to solve the systems of nonlinear Fredholm integro-diﬀerential equations of the second kind. The method approaches any Fredholm equation in a direct manner and gives the solution in an exact form and not in a series form. The method will be applied for the degenerate or separable kernels of the form n n ˜ k (t), ˜ K1 (x, t) = gk (x)hk (t), K1 (x, t) = g˜k (x)h K2 (x, t) =

k=1 n k=1

rk (x)sk (t),

˜ 2 (x, t) = K

k=1 n k=1

(16.113) r˜k (x)˜ sk (t).

536

16 Nonlinear Fredholm Integro-Diﬀerential Equations

The direct computation method can be applied as follows: 1. We ﬁrst substitute (16.113) into the system of Fredholm integrodiﬀerential equations (16.112) to obtain b b n n ˜ k (t)u(t)dt, u(i) (x) = f1 (x) + gk (x) hk (t)u(t)dt + g˜k (x) h k=1

v (i) (x) = f2 (x) +

n

a

k=1

b

rk (x)

sk (t)v(t) dt + a

k=1

n k=1

a b

r˜k (x)

s˜k (t)v(t) dt. a

(16.114) 2. Each integral at the right side depends only on the variable t with constant limits of integration for t. This means that Equation (16.114) becomes u(i) (x) = f1 (x) + α1 g1 (x) + · · · + αn gn (x) + β1 g˜1 (x) + · · · + βn g˜n (x), v (i) (x) = f2 (x) + γ1 r1 (x) + · · · + γn rn (x) + δ1 r˜1 (x) + · · · + δn r˜n (x), (16.115) where b b ˜ i (t) v(t)dt, 1 i n, hi (t) u(t)dt, 1 i n, βi = αi = h

a

b

si (t) u(t)dt, 1 i n,

γi = a

a b

s˜i (t) v(t)dt, 1 i n.

δi = a

(16.116) 3. Integrating both sides of (16.115) i times from 0 to x, and substituting the resulting equations for u(x) and v(x) into (16.116) gives a system of algebraic equations that can be solved to determine the constants αi , βi , γi , and δi . Using the obtained numerical values of these constants, the solutions u(x) and v(x) of the system (16.112) follow immediately. Example 16.17 Solve the system of nonlinear Fredholm integro-diﬀerential equations by using the direct computation method π 2

π3 u (x) = sin x + x cos x − u (t) + v 2 (t) dt, u(0) = 0 + 3 0 (16.117) π

2 π 2 u (t) − v (t) dt, v(0) = 0. v (x) = cos x − x sin x + + 2 0 Following the analysis presented above, this system can be rewritten as π3 , u (x) = sin x + x cos x + α + β − 3 (16.118) π , v (x) = cos x − x sin x + α − β + 2 where π π α= u2 (t) dt, β = v 2 (t) dt. (16.119) 0

0

16.4 Systems of Nonlinear Fredholm Integro-Diﬀerential Equations

537

Integrating both sides of (16.118) once from 0 to x gives π3 x, u(x) = x sin x + α + β − 3 (16.120) π x. v(x) = x cos x + α − β + 2 Substituting (16.120) into (16.119), and solving the resulting system gives 1 1 1 1 α = π3 − π, β = π3 + π. (16.121) 6 4 6 4 Substituting (16.121) into (16.120) leads to the exact solutions (u(x), v(x)) = (x sin x, x cos x).

(16.122)

Example 16.18 Solve the system of nonlinear Fredholm integro-diﬀerential equations by using the direct computation method ln 2 15 + (u2 (t) + v 2 (t))dt, u(0) = 1, u (x) = sinh x − 16 0 (16.123) v (x) = cosh x − ln 2 +

ln 2

0

(u2 (t) − v 2 (t))dt, v(0) = 0.

This system can be rewritten as

15 u (x) = sinh x + α + β − , 16 v (x) = cosh x + (α − β − ln 2),

where

α= 0

ln 2

u2 (t) dt,

β=

ln 2

v 2 (t) dt.

(16.124)

(16.125)

0

Integrating both sides of (16.124) once from 0 to x, and using the initial conditions we ﬁnd 15 x, u(x) = cosh x + α + β − 16 (16.126) v(x) = sinh x + (α − β − ln 2)x. To determine α, and β, we substitute (16.126) into (16.125) and solving the resulting system we obtain 15 1 15 1 + ln 2, β = − ln 2. (16.127) α= 32 2 32 2 Substituting (16.127) into (16.126) leads to the exact solutions (u(x), v(x)) = (cosh x, sinh x).

(16.128)

Example 16.19 Solve the system of nonlinear Fredholm integro-diﬀerential equations by using the direct computation method

538

16 Nonlinear Fredholm Integro-Diﬀerential Equations

u (x) = ex − 12 +

3x

v (x) = 3e

ln 2

(u2 (t) + v 2 (t))dt, u(0) = 1,

0

+9+ 0

ln 2

(16.129) 2

2

(u (t) − v (t))dt, v(0) = 1.

Proceeding as before we set u (x) = ex + (α + β − 12), v (x) = 3e3x + (α − β + 9), where ln 2

α=

u2 (t) dt,

0

ln 2

β=

v 2 (t) dt.

(16.130)

(16.131)

0

Integrating both sides of (16.130) once from 0 to x, and using the initial conditions we ﬁnd u(x) = ex + (α + β − 12)x, (16.132) v(x) = e3x + (α − β + 9)x. To determine α, and β, we substitute (16.132) into (16.131) and solving the resulting system we obtain 3 21 α= , β= . (16.133) 2 2 Substituting (16.133) into (16.132) leads to the exact solutions (u(x), v(x)) = (ex , e3x ).

(16.134)

Example 16.20 Solve the system of nonlinear Fredholm integro-diﬀerential equations by using the direct computation method π2 2 (v(t)w(t)) dt, u (x) = − sin x − + 3 0 u(0) = 0, u (0) = 1 π2 2 v (x) = − cos x − + (w(t)u(t)) dt, 3 0

(16.135)

v(0) = 1, v (0) = 0, π2 1 + (u(t)v(t)) dt, w (x) = −4 sin 2x − 2 0 w(0) = 0, w (0) = 2. This system can be rewritten as 2 , u (x) = − sin x + α − 3 2 , v (x) = − cos x + β − 3

(16.136)

16.4 Systems of Nonlinear Fredholm Integro-Diﬀerential Equations

w (x) = −4 sin 2x + γ − where

π 2

α=

1 2

539

,

v(t)w(t) dt,

0

β=

π 2

w(t)u(t) dt,

(16.137)

0

γ=

π 2

u(t)v(t) dt.

0

Integrating both sides of (16.136) twice from 0 to x, and using the initial conditions we obtain 1 1 α− x2 , u(x) = sin x + 2 3 1 1 β− x2 , v(x) = cos x + (16.138) 2 3 1 1 γ− x2 . w(x) = sin 2x + 2 4 Proceeding as before we obtain 2 1 2 (16.139) α= ,β= ,γ= . 3 3 2 This in turn gives the exact solutions (u(x), v(x), w(x)) = (sin x, cos x, sin 2x). (16.140)

Exercises 16.4.1 Use the direct computation method to solve the following systems of nonlinear Fredholm integro-diﬀerential equations π ⎧

⎪ 3 ⎪ 3 u2 (t) + v 2 (t) dt, u(0) = 0 ⎨ u (x) = cos x − x sin x − π + 0π 1.

2 π ⎪ ⎪ ⎩ v (x) = sin x + x cos x − + u (t) − v 2 (t) dt, v(0) = 0 2 0 ⎧ ln 2 2

15 1 ⎪ ⎪ u (x) = sinh 2x + (ln 2) + + u (t) − u(t)v(t) dt ⎪ ⎪ 2 32 ⎪ 0 ⎨ ln 2

2 3 15 2. ⎪ + v (t) − u(t)v(t) dt v (x) = sinh 2x − ln 2 − ⎪ ⎪ 2 32 ⎪ 0 ⎪ ⎩ u(0) = 1, v(0) = 2 π ⎧ π 4 ⎪ ⎪ (x) = − sin x − u2 (t) v 2 (t) dt, u(0) = 1 + u ⎨ 4 0 3. π ⎪ ⎪ ⎩ v (x) = sec x tan x − π + 2 u(t) v(t) dt, v(0) = 1 4 0

540

16 Nonlinear Fredholm Integro-Diﬀerential Equations

ln 2 ⎧

2 15 ⎪ ⎪ u (x) = sinh(2x) − (ln 2) − + u (t) + u(t)v(t) dt ⎪ ⎪ ⎪ 16 0 ⎨ ln 2

2 15 4. ⎪ + v (t) + u(t)v(t) dt v (x) = − sinh(2x) − 3 ln 2 + ⎪ ⎪ 16 ⎪ 0 ⎪ ⎩ u(0) = 1, v(0) = 1 ln 2 ⎧

x−2t 2 1 ⎪ ⎪ u (x) = (1 + x)ex − (ln 2)3 (ex − 1) + u (t) − u(t)v(t) dt, u(0) = 0 e ⎨ 3 0 5. ln 2 ⎪ x+2t 2

⎪ ⎩ v (x) = (1 − x)e−x − 1 (ln 2)3 (ex + 1) + e v (t) + u(t)v(t) dt, v(0) = 0 3 0 ⎧ π

π 2 ⎪ ⎪ u2 (t) + u(t)v(t) dt u (x) = − cos x − sin x − − 1 + ⎪ ⎪ 2 ⎪ 0 ⎨ π

π 6. 2 2 ⎪ v (t) + u(t)v(t) dt v (x) = − cos x + sin x − + 1 + ⎪ ⎪ 2 ⎪ 0 ⎪ ⎩ u(0) = 1, u (0) = 1, v(0) = 1, v (0) = −1 1 ⎧

2 1 ⎪ ⎪ u (x) = ex + − e2 + u (t) + v 2 (t) dt, u(0) = 1, u (0) = 2 ⎨ 3 0 7. 1

2 ⎪ ⎪ ⎩ v (x) = −ex − 4 + u (t) − v 2 (t) dt, v(0) = −1, v (0) = 0 0

ln 2 ⎧ 255 ⎪ x ⎪ (x) = e − v(t)w(t)dt, u(0) = 1, u (0) = 1 + u ⎪ ⎪ 8 ⎪ 0 ⎪ ⎪ ln 2 ⎨ 21 8. + v (x) = 9e3x − w(t)u(t)dt, v(0) = 1, v (0) = 3 ⎪ 2 ⎪ 0 ⎪ ln 2 ⎪ ⎪ ⎪ 15 ⎪ ⎩ w (x) = 25e5x − u(t)v(t)dt, w(0) = 1, w (0) = 5 + 4 0

16.4.2 The Variational Iteration Method The variational iteration method was used to handle the Fredholm integral equations, the Fredholm integro-diﬀerential equations and the nonlinear Fredholm integro-diﬀerential equations. The method provides rapidly convergent successive approximations of the exact solution if such a closed form solution exists, and not components as in Adomian decomposition method. The variational iteration method handles linear and nonlinear problems in the same manner without any need to speciﬁc restrictions such as the so called Adomian polynomials that we need for nonlinear terms. The correction functionals for the system of nonlinear Fredholm integrodiﬀerential equations b (i) % 1 (x, t)F%1 (v(t)) dt, K1 (x, t)F1 (u(t)) + K u (x) = f1 (x) + a (16.141) b % 2 (x, t)F%2 (v(t)) dt. K2 (x, t)F2 (u(t)) + K v(i) (x) = f2 (x) + a

16.4 Systems of Nonlinear Fredholm Integro-Diﬀerential Equations

are given by

x

0 x

un+1 (x) = un (x) + vn+1 (x) = vn (x) +

b

Γ1 =

λ(t) u(i) n (t) − f1 (t) − Γ1 (t) dt, λ(t)

0

vn(i) (t)

− f2 (t) − Γ2 (t) dt.

Γ2 =

(16.142)

% 1 (t, r)F%1 (˜ K1 (t, r)F1 (˜ un (r)) + K vn (r)) dr,

a b

541

% 2 (t, r)F%2 (˜ un (r)) + K vn (r)) dr. K2 (t, r)F2 (˜

(16.143)

a

As stated before it is necessary to determine the Lagrange multiplier λ that can be identiﬁed optimally. An iteration formula, without restricted variation, should be used for the determination of the successive approximations un (x), vn (x), n 0 of the solutions u(x) and v(x). The zeroth approximations u0 (x) and v0 (x) can be selected by using the initial conditions. Consequently, the solutions are given by u(x) = lim un (x), v(x) = lim vn (x). n→∞

n→∞

(16.144)

The VIM will be illustrated by studying the following examples. Example 16.21 Use the variational iteration method to solve the system of nonlinear Fredholm integro-diﬀerential equations π u (x) = cos x − 5 sin x + cos(x − t)(u2 (t) + v 2 (t))dt, u(0) = 1, 0

v (x) = 3 cos x − sin x +

0

π

sin(x − t)(u2 (t) + v 2 (t))dt, v(0) = 1. (16.145)

The correction functionals for this system are given by x un+1 (x) = un (x) − (un (t) − cos t + 5 sin t − ρ1 ) dt, vn+1 (x) = vn (x) −

0

where ρ1 =

ρ2 =

0 x

π 0

(16.146) (vn (t)

cos(t − r)(u2n (r) + vn2 (r)) dr, (16.147)

π 0

− 3 cos t + sin t − ρ2 ) dt,

sin(t −

r)(u2n (r)

+

vn2 (r)) dr.

We can select the zeroth approximations u0 (x) = u(0) = 1 and v0 (x) = v(0) = 1. Using this selection, the correction functionals give the following successive approxim